F Havet
email: fhavet@sophia.inria.fr

Choosability of the square of planar subcubic graphs with large girth

We show that the choice number of the square of a subcubic graph with maximum average degree less than 18/7 is at most 6. As a corollary, we get that the choice number of the square of a subcubic planar graph with girth at least 9 is at most 6. We then show that the choice number of the square of a subcubic planar graph with girth at least 13 is at most 5.

Introduction

Let G be a (simple) graph. The neighbourhood of a vertex v of G, denoted N G (v), is the set of its neighbours, i.e. is the set of vertices y such that xy is an edge. The degree of a vertex v in G, denoted d G (v), is its number of neighbours. Often, when the graph G is clearly understood from the context, we omit the subscript G. A graph is subcubic if every vertex has degree at most 3.

Let p : V (G) → IN. A p-list-assignment is a list-assignment L such that |L(v)| = p(v) for any v ∈ V (G). G is p-choosable if it is L-colourable for any p-list-assignment. By extension, if k is an integer, we say that G is k-choosable if it is p-choosable when p is the constant function with value k (i. e. p(v) = k for all v ∈ V ). The choice number of G, denoted ch(G), is the smallest integer k such that G is k-choosable. Clearly the choice number of G is at least as large as χ(G), the chromatic number of G.

The square of G is the graph G 2 with vertex set V (G) such that two vertices are linked by an edge of G 2 if and only if x and y are at distance at most 2 in G. A graph is called planar if it can be embedded in the plane. Wegner [?] proved that the square of a subcubic planar graph is 8-colourable. He also conjectured it is 7-colourable. Recently, this conjecture was proved by Thomassen [?].

Theorem 1 (Thomassen [?]) Let G be a subcubic planar graph. Then χ(G 2 ) ≤ 7. Kostochka and Woodall [?] conjectured that, for every square of a graph, the chromatic number equals the choice number.

Conjecture 2 (Kostochka and Woodall [?]) For all G, ch(G 2 ) = χ(G 2 ).

If true, this conjecture together with Theorem ?? implies that every subcubic planar graph is 7-choosable. Very recently, Cranston and Kim [?] showed that the square of every subcubic graph (non necessarily planar) other than the Petersen graph is 8-choosable.

The Hence the results of Dvořák, Škrekovski and Tancer imply that the choice number of the square of a planar graph with girth g is at most 6 if g ≥ 10, at most 5 if g ≥ 14 and at most 4 if g ≥ 24. The two latter results had been previously proved by Montassier and Raspaud [?].

average degree of G, denoted Ad(G) is v∈V (G) d(v) |V (G)| = 2|E(G)| |V (G)| .
In this paper, we improve some of these results. We first show (Theorem ??) that the choice number of the square of a subcubic graph with maximum average degree less than 18/7 is at most 6. As a corollary, we get that the choice number of the square of a subcubic planar graph with girth at least 9 is at most 6. Note that this corollary has been proved later and independently by Cranston and Kim [?]. We then show (Theorem ??) that the choice number of the square of a subcubic planar graph with girth at least 13 is at most 5.

The main results

The general frame of the proofs is classical. We consider a k-minimal graph, that is a subcubic graph such that its square is not k-choosable but the square of every proper subgraph is k-choosable. We prove that some configurations (i.e. induced subgraphs) are forbidden in such a graph and then deduce a contradiction. To do so, we will need the following definitions:

An i-vertex is a vertex of degree i. We denote by V i the set of i-vertices of G and by v i its cardinality. Let v be a vertex. An i-neighbour of v is a neighbour of v with degree i. The i-neighbourhood of v is

N i (v) = N (v) ∩ V i and its i-degree is d i (v) = |N i (v)|.
Some properties of 6-and 5-minimal graphs have already been proved in [?]. The easy first one is that V 0 ∪ V 1 = ∅, so G has minimum degree 2. This will allow us to use the following definitions for 6and 5-minimal graphs.

Let G be a subcubic graph with minimum degree 2. A thread of G is a path whose endvertices are 3-vertices and whose internal vertices are 2-vertices. The kernel of G is the weighted graph K G such that V (K G ) = V 3 (G) and xy is an edge in K G with weight l if and only if x and y are connected by a thread of length l in G. An edge of weight l is also called l-edge. Let x be a 3-vertex of G. The type of x is the triple (l 1 , l 2 , l 3 ) such that l 1 ≤ l 2 ≤ l 3 and the three edges (a loop being counted twice) incident to x have weight l 1 , l 2 and l 3 in K G . We denote by Y l1,l2,l3 the set of 3-vertices of type (l 1 , l 2 , l 3 ) and y l1,l2,l3 its cardinality. Moreover, for every integer i, we define Z i := l1+l2+l3=i Y l1,l2,l3 and z i = |Z i |. The number of vertices and edges and thus the average degree of G may be easily expressed in terms of the z i :

|V (G)| = i≥3 i -1 2 z i 2|E(G)| = i≥3 i.z i Ad(G) = i≥3 i.z i i≥3 i-1 2 z i (1)

6-choosability

The aim of this subsection is to prove the following result.

Theorem 4 Let G be a subcubic graph of maximum average degree d < 18/7. Then G 2 is 6-choosable.

Remark 5 Theorem ?? is tight. Indeed, the graph J 7 depicted in Figure ?? has average degree 18/7 and its square is the complete graph on seven vertices K 7 which is not 6-choosable (nor 6-colourable).

J 7 Theorem ?? and Proposition ?? yield that the square of a subcubic planar graph with girth at least 9 is 6-choosable.

Corollary 6

The square of a subcubic planar graph with girth at least 9 is 6-choosable.

In order to prove Theorem ??, we need to establish some properties of 6-minimal graphs. Some of them have been proved in [?].

Lemma 7 (Dvořák, Škrekovski and Tancer [?]) Let G be a 6-minimal graph. Then the following hold:

(i) all the edges of K G have weight at most 2;

(ii) every 3-cycle of G has its vertices in V 3 ;

(iii) every 4-cycle of G has at least three vertices in

V 3 ; (iv) a vertex of Y 2,2,2 is not adjacent in K G to a vertex of Y 1,2,2 ∪ Y 2,2,2 .
We will prove in Subsection ?? some new properties.

Lemma 8 Let G be a 6-minimal graph. Then the following hold:

(i) if (v 1 , v 2 , v 3 , v 4 , v 1 ) is a 4-cycle with v 2 ∈ V 2 then v 1 or v 3 is not in Y 1,2,2 ; (ii) a vertex of Y 1,2,2 is adjacent in K G to at most one vertex of Y 1,2,2 by 2-edges.
Proof of Theorem ??. Let G be a 6-minimal planar graph. G has minimum degree 2, so its kernel K G is defined. Moreover by Lemma ?? (i), Z i is empty for i ≥ 7 and

Z 6 = Y 2,2,2 and Z 5 = Y 1,2,2 .
Let us consider a vertex of

Z 4 = Y 1,1,2 . Its neighbour in K G via the 2-edge is in Z 4 ∪ Z 5 ∪ Z 6 because a vertex of Z 3 = Y 1,

5-choosability

Dvořák, Škrekovski and Tancer [?] proved that the square of a subcubic graph G with maximum average degree less than 7/3 is 5-choosable. This result is tight since the graph J 6 depicted in Figure ?? has average degree 7/3 and its square is the complete graph on six vertices K 6 which is not 5-choosable (nor 5-colourable). However, we will prove that the square of a subcubic planar graph with girth at least 13 J 6

Figure 2: The graph J 6 is 5-choosable, which improves the result of Montassier and Raspaud.

Theorem 9

The square of a subcubic planar graph with girth at least 13 is 5-choosable.

In order to prove this theorem, we need to establish some properties of 5-minimal graphs. Some of them have been proved in [?].

Lemma 10 (Dvořák, Škrekovski and Tancer [?]) Let G be a 5-minimal graph. Then the following hold:

(i) all the edges of K G have weight at most 3;

(ii) if i ≥ 8, Z i is empty.
We will prove in Subsection ?? some new properties.

Lemma 11 Let G be a 5-minimal graph of girth at least 13. Then in K G the following hold: Proof of Theorem ??. Let G be a 5-minimal planar graph with girth at least 13. G has minimum degree 2, so its kernel K G is defined. Moreover, by Lemma ?? (i), Z 7 = Y 2,2,3 ∪ Y 1,3,3 , so

z 7 = y 2,2,3 + y 1,3,3 . (2)
Let us count the number e 2 of 2-edges incident to vertices of Y 2,2,3 . Recall that

Z 4 = Y 1,1,2 and Z 3 = Y 1,1,1
Let us now count the number e 1 of 1-edges incident to vertices of Y 1,3,3 . Since 1-edges may not link two vertices of type [START_REF] Borodin | Criterion of chromaticity of a degree prescription[END_REF][START_REF] Dvořák | List-colouring squares of sparse subcubic graphs[END_REF][START_REF] Dvořák | List-colouring squares of sparse subcubic graphs[END_REF] according to Lemma ?? (ii), we have e 1 = y 1,3,3 . Moreover, the ends of such edges which are not in Y 1,1,3 have to be in Y 1,2,2 ∪Y 1,1,3 ∪Z 4 ∪Z 3 by Lemmas ?? and ?? (ii). Furthermore, vertices of Y 1,2,2 (resp. Y 1,1,3 ∪ Z 4 , Z 3 ) are incident to at most one (resp. two, three) 1-edges. Thus e 1 ≤ y 1,2,2 + 2y 1,1,3 + 2z 4 + 3z 3 . So,

y 1,3,3 ≤ y 1,2,2 + 2y 1,1,3 + 2z 4 + 3z 3 . (4)
2 × (??) + (??) yields 2y 2,2,3 + 2y This contradicts the fact that G has girth 13 by Proposition ??.

Remark 12 It is very likely that using the method below, one can prove that a graph G with maximum average degree less than 26 11 is 5-choosable unless it contains J 6 as an induced subgraph. However, this will require the tedious study of a large number of configurations.

3 Proofs of Lemmas ?? and ??

In order to prove Lemmas ?? and ??, we need the following lemma proved in

G. If (G -S) 2 = G 2 -S, then G 2 [S] is not p S -choosable.
In order to use Lemma ??, we need some results on the choosability of some graphs.

Some choosability tools

Definition 14 Let x and y be two vertices of a graph G. An (xy)-ordering of G is an ordering of the vertices such that x is the minimum and y the maximum. An (x, yz)-ordering is an ordering of the vertices such that x is minimum, y is the second minimum and z is maximum.

Let σ = (v 1 < v 2 < . . . < v n ) be an ordering of the vertices of G and p a function

V (G) → IN. σ is p-greedy if, for every i, |N (v i ) ∩ {v 1 , . . . , v i-1 }| < p(v i ). It is p-nice if, for every i except n, |N (v i ) ∩ {v 1 , . . . , v i-1 }| < p(v i ) and d(v n ) = p(v n ). It is p-good if, for every 3 ≤ i ≤ n, |N (v i ) ∩ {v 1 , . . . , v i-1 }| -ǫ(v i ) < p(v i ) with ǫ(v i ) = 1 if v i is adjacent to both v 1 and v 2 and ǫ(v i ) = 0 otherwise. By extension, if k is an integer, we say that σ is k-greedy (resp. k-nice, k-good) if it is p-greedy (resp. p-nice, p-good) when p is the constant function with value k (i. e. p(v i ) = k for every 1 ≤ i ≤ n).
The greedy algorithm according to greedy, nice and good orderings yields the following three lemmas.

Lemma 15 If G has a p-greedy ordering then G is p-choosable.

Proof. Applying the greedy algorithm according to the p-greedy ordering gives the desired colouring.

Lemma 16 Let xy be an edge of graph G and L be a p-list-assignment of G. If L(x) ⊂ L(y) and G has a p-nice (xy)-ordering, then G is L-colourable.

Proof. Let a be a colour in L(x) \ L(y). Assign a to x and proceed the greedy algorithm according to the p-nice (xy)-ordering.. The only vertex which has not more colour in its list than previously coloured neighbours is y for which |L(y)| = d(y). But since a / ∈ L(y), at most d(y) -1 colours of L(y) are assigned to the neighbours of y. Hence one can colour y.

Lemma 17 Let x, y and z be three vertices of a graph

G = (V, E) such that xy / ∈ E. If L(x) ∩ L(y) = ∅ and G has a p-good (x, y -z)-ordering, then G is L-colourable.
Proof. Let a be a colour in L(x)∩L(y) and σ = (v 1 < v 2 < . . . < v n ) be a p-good (x, yz)-ordering. (In particular, v 1 = x, v 2 = y and v n = z.) Assign a to x and y and proceed the greedy algorithm according to σ. For every 3 ≤ i ≤ n, the number of colours assigned to already coloured neighbours of v i is at most |N (v i ) ∩ {v 1 , . . . , v i-1 }|ǫ(v i ) since v 1 and v 2 are coloured the same. Hence the greedy algorithm gives an L-colouring.

Remark 18 Note that if xz, yz ∈ E, a p-nice (x, yz)-ordering is also p-good.

Definition 19

The blocks of a graph are its maximal 2-connected components. A connected graph is said to be a Gallai tree if each of its blocks is either a complete graph or an odd cycle.

The following theorem was proved independently by Borodin [?] and Erdős, Rubin and Taylor [?]:

Theorem 20 (Borodin [?], Erdős, Rubin and Taylor [?]) Let G be a connected graph and d G the degree function in G. Then G is d G -choosable if and only if G is not a Gallai tree.

Lemma 21 Let G = (V, E) be a graph and p :

V (G) → IN. Let S be a set of vertices such that p(v) ≥ d(v) for all v ∈ S. If G[S] is not a Gallai tree and G -S is p-choosable then G is p-choosable.
Proof. Let L be a p-list-assignment of G. Since G -S is p-choosable, its admits an L-colouring c. Let us now extend it to S. The list

I(v) = L(v) \ {c(w), w ∈ N (v) \ S} of available colours of a vertex v ∈ S is of size at least p ′ (v) = p(v) -|N (v) \ S| ≥ d G[S] (v). Since G[S]
is not a Gallai tree, by Theorem ??, G[S] is p ′ -choosable and thus I-colourable. So, G is L-colourable.

A 4-regular graph G is cycle+triangles if it is the edge union of a Hamiltonian cycle C and a 2-factor consisting of triangles. In other words, the graph induced by the edges of E(G) \ E(C) is the disjoint union of 3-cycles.

Theorem 22 (Fleischner and Stiebitz [?]) Every cycle+triangles graph is 3-choosable.

Proof of Lemma ??

Lemma 23 Let q ≥ 2 and C 4q = (v 1 , . . . , v 4q , v 1 ) be the 4q-cycle and p defined by p(v i ) = 4 if i is odd and p(v i ) = 2 otherwise. Then C 2 4q is p-choosable.

Proof. The set S of vertices v for which p(v) ≥ d C 2 4q (v) is the set of v i with odd indices. C 2 4q [S] is a 2q-cycle and thus is not a Gallai tree. Moreover C 2 4q -S is also a 2q-cycle and is 2-choosable. Hence Lemma ?? gives the result.

Proposition 24 Let P 7 = (v 1 , . . . , v 7 ) be a path and p the function defined by p(v 1 ) = p(v 2 ) = p(v 6 ) = p(v 7 ) = 2, p(v 3 ) = p(v 5 ) = 4 and p(v 4 ) = 3. Then P 2 7 is p-choosable.

Proof. Let L be a p-list-assignment of

P 2 7 . Since (v 2 < v 4 < v 6 < v 7 < v 5 < v 3 < v 1
) is a p-nice ordering of P 2 7 , by Lemma ??, we may assume that L(v 1 ) = L(v 2 ), and by symmetry of P 7 and p that

L(v 6 ) = L(v 7 ). Since (v 1 < v 4 < v 2 < v 6 < v 7 < v 5 < v 3
) is p-good, by Lemma ??, we may assume that L(v 1 ) ∩ L(v 4 ) = ∅, and by symmetry

L(v 7 ) ∩ L(v 4 ) = ∅. Now one can find c(v 1 ) ∈ L(v 1 ), c(v 2 ) in L(v 2 ) \ {c(v 1 )}, c(v 6 ) in L(v 6 ), c(v 7 ) in L(v 7 ) \ {c(v 6 )}, c(v 3 ) in L(v 3 ) \ {c(v 1 ), c(v 2 )}, and c(v 5 ) in L(v 5 ) \ {c(v 3 ), c(v 6 ), c(v 7 )}. Now since L(v 1 ) ∩ L(v 4 ) = ∅ and L(v 1 ) = L(v 2 ), c(v 2 ) / ∈ L(v 4 ). Analogously, c(v 6 ) / ∈ L(v 4 ). Hence, L(v 4 ) \ {c(v 2 ), c(v 3 ), c(v 5 ), c(v 6 )} = L(v 4 ) \ {c(v 3 ), c(v 5 )} = ∅.
So, one can choose c(v 4 ) in this set to get an L-colouring c of P 2 7 .

Lemma 25 For 1 ≤ i ≤ 17, let F i be the graphs and p i be the functions depicted in Figure ??.

(i) F 2 1 ∪ {v 5 v 6 } is p 1 -choosable. (ii) F 2 2 ∪ {v 1 v 4 } and F 2 2 ∪ {v 4 v 7 } are p 2 -choosable. (iii) F 2 3 ∪ {v 4 v 8 } is p 3 -choosable. (iv) F 2 4 is 6-choosable. (v) F 2 5 ∪ {v 1 v 4 , v 1 v 6 } is p 5 -choosable. (vi) F 2 6 is p 6 -choosable. (vii) F 2 7 ∪ {v 9 v 10 } is p 7 -choosable. (viii) F 2 8 is p 8 -choosable. (ix) F 2 9 ∪ {v 2 v 9 }
v 8 , v 8 v 9 , v 9 v 4 } is 5-choosable. (xii) F 2 12 ∪ {v 4 v 8 } is p 12 -choosable. (xiii) F 2 13 is 6-choosable. Proof. (i) In F 2 1 ∪ {v 5 v 6 }, (v 6 < v 5 < v 4 < v 3 < v 1 < v 2 ) is p 1 -greedy. So, by Lemma ??, F 2 1 ∪ {v 5 v 6 } is p 1 -choosable. (ii) In F 2 2 ∪ {v 4 v 7 }, (v 2 < v 4 < v 7 < v 6 < v 5 < v 3 < v 1 ) is p 2 -nice and p 2 (v 2 ) > p 2 (v 1 ). So, by Lemma ??, F 2 2 ∪ {v 4 v 7 } is p 2 -choosable. By symmetry, one shows that F 2 2 ∪ {v 1 v 4 } is p 2 -choosable. (iii) In F 2 3 ∪ {v 4 v 8 }, (v 2 < v 8 < v 4 < v 7 < v 6 < v 5 < v 3 < v 1
) is p 3 -nice and p 3 (v 2 ) > p 3 (v 1 ). So, by Lemma ??, F 2 3 ∪ {v 4 v 8 } is p 3 -choosable.

(iv) Let L be a 6-list-assignment of F 2 4 . Every ordering with maximum v 1 and second maximum v 7 is 6nice. Thus, by Lemma ??, we may assume that L(v j ) = L(v 1 ) for j ∈ {2, 3, 4, 5, 6, 8}. Analogously, we may assume that L(v j ) = L(v 7 ) for j ∈ {2, 3, 4, 5, 6, 8}. Hence all the lists are the same, say {1, 2, 3, 4, 5, 6}. Now c(

v 1 ) = c(v 5 ) = 1, c(v 2 ) = 2, c(v 3 ) = c(v 7 ) = 3, c(v 4 ) = 4, c(v 6 ) = 5 and c(v 8 ) = 6 is an L-colouring of F 2 4 . (v) In F 2 5 ∪ {v 1 v 4 , v 1 v 6 }, (v 7 < v 6 < v 1 < v 4 < v 2 < v 3 < v 5
) is p 5 -nice and p 5 (v 7 ) > p 5 (v 5 ). So, by Lemma ??,

F 2 5 ∪ {v 1 v 5 } is p 5 -choosable. (vi) In F 2 6 , (v 4 < v 2 < v 8 < v 1 < v 3 < v 5
) is p 6 -nice and p 6 (v 4 ) > p 6 (v 5 ). So, by Lemma ??, F 2 6 is p 6 -choosable.

(vii) Let L be a p 7 -list-assignment of F 2 7 ∪ {v 9 v 10 }. (v 2 < v 9 < v 10 < v 8 < v 6 < v 4 < v 7 < v 5 < v 3 < v 1 ), (v 2 < v 9 < v 10 < v 8 < v 6 < v 4 < v 5 < v 7 < v 1 < v 3 ) and (v 4 < v 9 < v 10 < v 8 < v 6 < v 2 < v 5 < v 7 < v 1 < v 3 ) are p 7 -nice. Thus, by Lemma ??, we may assume that L(v 2 ) ⊂ L(v 1 ), L(v 2 ) ⊂ L(v 3 ) and L(v 4 ) ⊂ L(v 3 ). It follows that L(v 1 ) ∩ L(v 4 ) = ∅. Because (v 1 < v 4 < v 10 < v 9 < v 2 < v 8 < v 6 < v 7 < v 5 < v 3 ) is p 7 -good, by Lemma ??, F 2 7 ∪ {v 9 v 10 } is L-colourable.

(viii) In F 2 8 , (v 6 < v 5 < v 7 < v 9 < v 8 < v 4 < v 3 < v 2 < v 1 ) is p 8 -greedy. So, by Lemma ??, F 2 8 is p 8 -choosable.

Figure 1 :

 1 Figure 1: The graph J 7

  (i) a vertex of Y 2,2,3 and a vertex of Y 1,2,3 ∪ Y 2,2,3 are not linked by a 2-edge; (ii) a vertex of Y 1,3,3 and a vertex of Y 1,2,3 ∪ Y 1,3,3 are not linked by a 1-edge; (iii) s vertex of Y 2,2,2 is not adjacent in K G to three vertices of Y 2,2,3 (by 2-edges).

  [?]. Let S be a set of vertices of a k-minimal graph G. The function p S : S → IN is defined by p S (v) = k -|N G 2 (v) \ S|. Then p S (v) represents the minimum number of available colours at a vertex v ∈ S once we have precoloured the square of G -S. Hence if (G -S) 2 is k-choosable, (G -S) 2 = G 2 -S and G 2 [S] is p S -choosable, one can extend any k-list-colouring of G -S into a k-list-colouring of G, which is a contradiction. Lemma 13 (Dvořák, Škrekovski and Tancer [?]) Let S be a set of vertices of a k-minimal graph

Figure 3 :

 3 Figure 3: The graphs F i and functions p i for 1 ≤ i ≤ 13

  In [?], Dvořák, Škrekovski and Tancer proved that the choice number of the square of a subcubic graph G is at most 4 if M ad(G) < 24/11 and G has no 5-cycle, at most 5 if M ad(G) < 7/3 and at most 6 if M ad(G) < 5/2.The girth of a graph is the smallest length of a cycle in G. Planar graphs with prescribed girth have bounded maximum average degree: Proposition 3 Every planar graph with girth at least g has maximum average degree less than 2 + 4 g-2 .

The maximum average degree of G, denoted M ad(G), is max{Ad(H), H subgraph of G}.

  1,1 is incident to no edge of weight 2. For i = 4, 5, 6, let Z i 4 be the set of vertices of 3 Z 4 which are adjacent to a vertex of Z i by their unique 2-edge and z i 4 its cardinality. (Z 4 4 , Z 5 4 , Z 6 4 ) is a partition of Z 4 so z 4 = z4 4 + z 5 4 + z 6 4 . Hence Equation (??) becomes (iv), the three neighbours in K G of a vertex of Z 6 are not in Z 6 ∪ Z 5 .

	Ad(G) =	6z 6 + 5z 5 + 4z 6 4 + 4z 5 4 + 4z 4 4 + 3z 3 5 2 z 6 + 2z 5 + 3 2 z 6 4 + 3 2 z 5 4 + 3 2 z 4 4 + z 3	.
	By Lemma ?? So they must be
	in Z 6 4 . It follows that 3z 6 = z 6 4 . So
	Ad(G) =	5z 5 + 6z 6 4 + 4z 5 4 + 4z 4 4 + 3z 3 2z 5 + 7 3 z 6 4 + 3 2 z 5 4 + 3 2 z 4 4 + z 3	.
	By Lemma ?? (ii), a vertex of Z 5 is adjacent to at least one vertex of Z 5 4 . Thus z 5 ≤ z 5 4 . But Ad(G) is decreasing as a function of z 5 since z 6 4 , z 5 4 , z 4 4 and z 3 are non-negative. It follows that
	Ad(G) ≥	6z 6 4 + 9z 5 4 + 4z 4 4 + 3z 3 7 3 z 6 4 + 7 2 z 5 4 + 3 2 z 4 4 + z 3	≥	18 7	.

  . Since 2-edges may not link two vertices of type (2, 2, 3) according to Lemma ?? (i), we have e 2 = 2y 2,2,3 . Moreover, the ends of such edges which are not in Y 2,2,3 have to be in Y 2,2,2 ∪ Y 1,2,2 ∪ Z 4 by Lemmas ?? and ?? (i). Furthermore, a vertex of Y 2,2,2 is incident to at most two edges of e 2 according to Lemma ?? (iii) and a vertex of Y 1,2,2 (resp. Z 4 ) is incident to at most two (resp. one) 2-edges. Therefore e 2 ≤ 2y 2,2,2 + 2y 1,2,2 + z 4 . So, 2y 2,2,3 ≤ 2y 2,2,2 + 2y 1,2,2 + z 4 .

  1,3,3 ≤ 2y 2,2,2 + 4y 1,2,2 + 4y 1,1,3 + 5z 4 + 6z 3 . Hence, by Equation (??), 2z 7 ≤ 2z 6 + 4z 5 + 5z 4 + 6z 3 , so

	z 7 ≤ z 6 + 2z 5 +	5 2	z 4 + 3z 3 .
	Now by Equation (??) the average degree of G is
	Ad(G) =	7z 7 + 6z 6 + 5z 5 + 4z 4 + 3z 3 3z 7 + 5 2 z 6 + 2z 5 + 3 2 z 4 + z 3
	Ad(G) ≥	13z 6 + 19z 5 + 43 2 z 4 + 24z 3 11 2 z 6 + 8z 5 + 9z 4 + 10z 3	≥	26 11	.

.

As a function of z 7 , this is a decreasing function (on IR + ); so it is minimum when z 7 is maximum that is equal to z 6 + 2z 5 + 5 2 z 4 + 3z 3 . So,

  and F 2 9 ∪ {v 6 v 9 } are p 9 -choosable. (x) F 2 10 ∪ {v 4 v 8 } is p 10 -choosable. (xi) F 2 11 ∪{v 4 v 8 , v 8 v 9 }, F 2 11 ∪{v 4 v 8 , v 9 v 4 } and F 2 11 ∪{v 8 v 9 , v 9 v 4 } are p 11 -choosable and F 2 11 ∪{v 4

Acknowledgement

The author would like to thank André Raspaud and Riste Škrekovski for stimulating discussions. He is also grateful to the two anonymous referees whose comments helped a lot to improve the presentation of this paper.

* Partially supported by the European Project IST AEOLUS

(ix) Let L be a p 9 -list-assignment of F 2 9 ∪ {v 2 v 9 }. Then (v 2 < v 9 < v 6 < v 4 < v 8 < v 7 < v 5 < v 3 < v 1 ) and (v 2 < v 9 < v 6 < v 4 < v 8 < v 7 < v 5 < v 1 < v 3 ) are p 9 -nice so by Lemma ??, we may assume that L(v 2 ) ⊂ L(v 3 ) ∩ L(v 1 ). Moreover, (v 4 < v 2 < v 9 < v 6 < v 8 < v 7 < v 5 < v 1 < v 3 ) is p 9 -nice so by Lemma ??, we may assume that L(v 4 ) = L(v 3 ). It follows that L(v 1 ) ∩ L(v 4 ) = ∅. Thus, by Lemma ??, since (v

By symmetry, one shows that F 2 9 ∪ {v 6 v 9 } is p 9 -choosable.

This gives an L-colouring of F . So we may assume that L(v 8 ) ⊂ L(v 2 ). Exchanging the role of c 4 in c 8 in the preceding argument, we may assume that L(v 4 ) ⊂ L(v 2 ). Moreover by symmetry, we may assume that L(v 9 ) ∪ L(v 4 ) ⊂ L(v 6 ). In particular, this implies that the sets

) is p 11 -good, so by Lemma ??, F is L-colourable.

) is p 11 -good, so by Lemma ??, F is L-colourable.

) is p 11 -good, so by Lemma ??, F is L-colourable. (xiii) Let L be a 6-list-assignment of

Proof of Lemma ??.

To prove this lemma, we will suppose for a contradiction that it does not hold. Then we will find a set X of vertices contradicting Lemma ??. Indeed Lemma ?? will show that G 2 [X] is p X -choosable and for each set X we consider, every vertex of X has at most one neighbour in G -X, so (G -X) 2 = G 2 -X. Lemma ?? completes the proof.

(i) Suppose for a contradiction that v 1 and v 3 are in Y 1,2,2 . Let v 5 (resp. v 6 ) be the neighbour of v 1 (resp. v 3 ) distinct from v 2 and v 4 . By Lemma ??

So Lemma ?? contradicts Lemma ??.

(ii) Suppose for a contradiction that, in K G , a vertex v 4 of Y 1,2,2 is adjacent to two vertices of Y 1,2,2 v 2 and v 6 by 2-edges. According to Lemma ?? (iii), v 2 = v 6 . Let v 3 and v 5 be the 2-neighbours of v 4 common with v 2 and v 6 respectively, and v 1 (resp. v 7 ) be the 2-neighbour of v 2 (resp. v 6 ) not adjacent to v 4 . Set S = {v 1 , . . . , v 7 }.

We first claim that

a cycle+triangle graph, thus, by Theorem ??, it is 3-choosable and so p S -choosable. This contradicts Lemma ??.

w 1 (resp. w 7 ) be the neighbour of v 1 (resp. w 7 ) distinct from v 2 (resp. v 6 ) and for i ∈ {2, 4, 6}, let w i be the neighbour of

We claim that W ∩ S = ∅. Indeed, suppose for a contradiction that W ∩ S = ∅. Since G is simple, w 1 = v 2 and w 7 = v 6 . Moreover by Lemma ?? (i), w 1 and w 7 are in V 3 , so w 1 = v 7 and w 7 = v 1 . Furthermore, by Lemma ?? (ii), w 2 = v 4 and w 6 = v 4 and by Lemma ?? (iii), w 1 = v 4 and w 7 = v 4 . Last, we may not have w 1 = v 6 and w 2 = v 7 otherwise the 4-cycle (v 1 , v 6 , v 7 , v 2 , v 1 ) would contradict Lemma ?? (iii). Then, by symmetry, we only need to consider the cases

Thus, by Lemmas ?? and ??,

so G is 6-choosable, by Lemma ??. This is a contradiction.

Thus Lemma ?? contradicts Lemma ??.

This proves the claim.

Note that by Lemma ?? (ii), w 1 = w 2 and w 6 = w 7 . Suppose

. Thus Lemma ?? contradicts Lemma ??. Therefore, w 1 = w 4 and, by symmetry, w 4 = w 7 .

Suppose w 1 = w 7 = v 8 . Let T = S ∪ {v 8 }. Then G[T ] is the cycle C 8 and p T is greater or equal to the function p defined in Lemma ??. So, by Lemmas ?? and ??, G 2 [T ] = C 2 8 . It follows that either w 2 = w 6 or w 4 = w 8 with w 8 be the neighbour of v 8 not in S.

• Suppose w 2 = w 6 = v 9 , and

7 ∪ {v 9 v 10 }; so Lemma ?? contradicts Lemma ??. If not, G = G[W ] = F 13 , so G 2 is 6-choosable, according to Lemma ??, a contradiction.

. Hence Lemma ?? contradicts Lemma ??. By symmetry, we get a contradiction if w 2 = w 6 = w 8 , w 2 = w 4 = w 8 or w 4 = w 6 = w 8 .

• Suppose w 4 = w 8 = v 9 , w 2 = v 9 , w 6 = v 9 and w 2 = w 6 . Setting U = {v 1 , . . . , v 9 }, we have

Hence Lemma ?? contradicts Lemma ??.

By symmetry, we get a contradiction if w 2 = w 6 = v 9 , w 4 = v 9 , w 8 = v 9 and w 4 = w 8 . Therefore, w 1 = w 7 .

Suppose that w 2 = w 6 = v 8 . Let T = S ∪{v 8 }. Then G[T ] = F 10 , p T ≥ p 10 , and G 2 [T ] ⊂ F 2 10 ∪{v 4 v 8 }, since w 1 , w 4 and w 7 are distinct vertices. Hence Lemma ?? contradicts Lemma ??. Therefore, w 2 = w 6 .

Suppose that w 1 = w 6 = v 8 and

. Hence Lemma ?? contradicts Lemma ??.

Therefore, w 1 = w 6 or w 2 = w 7 . By symmetry, w 2 = w 7 . Suppose w 1 = w 6 = v 8 . Let T = S ∪ {v 8 } and let w 8 be the neighbour of v 8 not in S. Then

Hence Lemma ?? contradicts Lemma ??. Therefore, w 1 = w 6 .

Hence all the w i are distinct, so

. Thus Proposition ?? contradicts Lemma ??.

Remark 26 The proof of Lemma ?? in the case of planar graphs of girth at least 9 is simpler and shorter because all the configurations considered in the above proof (except the path P 7 ) have girth less than 9. Thus Corollary ?? has a short direct proof which requires only Proposition ??.

Proof of Lemma ??

Definition 27 For 1 ≤ j ≤ 4, let I j and q j be the graphs and functions depicted in Figure ??. Figure 4: The graphs I j and functions q j , 1 ≤ j ≤ 4

Lemma 28 For 1 ≤ j ≤ 4, I 2 j is q j -choosable.

Proof.

• Let L be a q 1 -list-assignment of

) are q 1 -nice. So, by Lemma ??, we may assume that L(v

) is q 1 -good. Thus, by Lemma ??,

Suppose now that L(v 4 ) ∩ L(v 6 ) = ∅. Then colour v 4 and v 6 with the same colour c(v

good so, by Lemma ??, we may assume that

nice so, by Lemma ??, we may assume that L(v 4 ) ⊂ L(v 2 ). Then one may assign c(v 4 ) ∈ L(v 4 ) and c(v 2 ) ∈ L(v 4 ) \ {c(v 4 )} to the vertices v 4 and v 2 . Now, because L(v 4 ) ∩ L(v 3 ) = ∅, one can extend c into an L-colouring of I 2 2 by colouring greedily according to the ordering (v

)) and to v 6 a colour in L(v 6 ) \ (L(v 8 ) ∪ {c 5 }). Then colour the remaining vertices greedily according to

. Pick c(y 1 ) in L(y 1 )\L(w 1 ), c(y 2 ) in L(y 2 )\(L(w 2 )∪{c(y 1 )}), c(y 3 ) in L(y 3 ) \ (L(w 3 ) ∪ {c(y 1 ), c(y 2 )}) and c(x) in L(x) \ {c(y 1 ), c(y 2 ), c(y 3 )}. Since I 2 1 is q 1 -choosable, one can extend c to a colouring of I 2 4 .

Proof of Lemma ??. (iii) Suppose that a vertex x of Y 2,2,2 is adjacent to three vertices v 1 , v 2 and v 3 of Y 2,2,3 in K G . Then the subgraph of G induced by x, v 1 , v 2 , v 3 and the 2-vertices of their incident threads is I 4 . So Lemma ?? contradicts Lemma ??.


Fernando Lopes

Michael Atlan

Singular value demodulation of phase-shifted holograms

We report on phase-shifted holographic interferogram demodulation by singular value decomposition. Numerical processing of optically-acquired interferograms over several modulation periods was performed in two steps : 1-rendering of off-axis complex-valued holograms by Fresnel transformation of the interferograms; 2-eigenvalue spectrum assessment of the lag-covariance matrix of hologram pixels. Experimental results in low-light recording conditions were compared with demodulation by Fourier analysis, in the presence of random phase drifts.

Recording optical holograms by time-modulated interferometry was proposed and demonstrated in early studies [1][2][START_REF] Dändliker | Progress in optics[END_REF]. It pioneered the development of phaseshifting [START_REF] Yamaguchi | [END_REF] holographic interferometry. Time-modulated phase-shifting interferometry involves recording of a sequence of interferograms over at least one modulation cycle of the intensity. Temporal demodulation consists in a linear combination of a sequence of recorded interferograms, whose coefficients are the ones of a discrete Fourier transform [5][6][7]. An important feature of holographic interferometry is to allow spatial signal modulation in off-axis recording conditions [8], which permits the discrimination of self-beating and crossbeating interferometric contributions of the object field and the reference field. In this configuration, spurious interferometric contributions can be filtered-off [9]. It was demonstrated that the combination of spatial and temporal signal modulation enables low-light imaging at shot noise levels [10,11]. Important features and applications include optical phase imaging [12][13][14][15] and detection tunability in the radiofrequency range for optical heterodyne detection [16], frequency-resolved narrowband sensing of scattered optical radiation [17][18][19]. Sensitivity is compatible with photon-counting recording conditions [20][21][22]. In standard phase-shifting interferometry, a sequence of interferograms with known phase shifts between probed and reference optical waves is recorded. When the modulation is not known, principal component analysis [23] can be used for nonparametric, non-iterative demodulation. It identifies uncorrelated variables in the structure of the data (the principal components) from the analysis of its correlations. Non-parametric approaches were also applied to image formation concepts [24], speckle reduction [25][26][27][28][29], and laser speckle contrast analysis of flows [START_REF] Humeau-Heurtier | Medical Imaging[END_REF].

In this letter, we demonstrate experimentally that the sensitivity of low-light optical signal retrieval in time-modulated holographic interferometry can be im-proved with respect to Fourier transform demodulation in practical conditions, when random signal fluctuations are present. For that purpose, a data-driven, nonparametric signal processing method is used. It consists of calculating the spectrum of eigenvalues in a singular value decomposition (SVD) of the lag-covariance matrix of recorded temporal fluctuations of the hologram's pixels, after spatial Fresnel transformation. The proposed processing scheme is less affected by random optical phase drifts occurring during the sampling process than signal demodulation by discrete Fourier transformation.

In optical heterodyne detection, a field of interest E(t) = E exp (iω L t), oscillating at the optical frequency ω L , is non-linearly mixed with a local oscillator field E LO (t) = E LO exp (i∆ωt) exp (iω L t) that is set at a closeby intermediate frequency ω L + ∆ω. In this notation, E and E LO are complex constants and i is the imaginary unit. The amplitude and phase of the original signal E(t) can be retrieved in the desired outcome oscillating at the difference frequency ∆ω, which can be set within the receiver's temporal bandwidth. The frequency conversion process is ensured by non-linear detection of the optical field E by the array of square-law sensors of a camera, which respond quadratically with the impinging electric field's magnitude. The squared magnitude of the total field received,

I(t) = |E(t) + E LO (t)| 2 ,
has cross-terms oscillating at the difference frequency ∆ω of the fields E and E LO , causing modulation of the optical energy flux

I(t) = |E| 2 + |E LO | 2 + EE * LO e -i∆ωt + E * E LO e i∆ωt
, where * denotes the complex conjugate. This equation describes the temporal fluctuation of the recorded intensity at a given pixel. Holographic image rendering, referred to as spatial demodulation, is then performed onto each recorded interferogram I by a discrete Fresnel transform. This transformation involves the free-space propagation impulse response of the optical field [7,31]. In off-axis recording configuration, the Fresnel transform separates spatially the four interferometric terms of I(t) [8,9] and forms a reconstructed hologram. In the absence of random fluctuations, the pixels of the off-axis part of the hologram carry the signal

X(t) = EE * LO exp(-i∆ωt) (1)
The purpose of signal demodulation is to retrieve the magnitude of the oscillating signal at the frequency ∆ω at each pixel. In ideal conditions, Eq. 1 holds and signal demodulation by discrete Fourier analysis of the recorded interferograms [6,7] enables shot-noise limited detection in dim light [10]; yet in actual experiments, random amplitude and phase fluctuations limit the detection sensitivity.

We recorded phase-shifted interferograms over several modulation periods in low-light and studied signal retrieval procedures. Optical recording of interferograms was performed with a frequency-shifted, off-axis Mach-Zehnder interferometer [32] to G H ≃ 5.5 × 10 6 , because the LO power was kept constant throughout the experiment. The holographic images of these two datasets were then demodulated spatially [31], to yield 661 × 661 complex-valued time traces X(t) of 128 elements each, cropped from the 2048 × 2048 calculation array, over an off-axis spatial region of interest where the image was expected. We then compared two types of signal demodulation procedures to assess the magnitude of the oscillating signal at the frequency ∆ω in X(t) at each pixel : Fourier analysis, and singular value analysis of temporal fluctuations. First, holographic images were rendered without demodulation. The squared magnitude of the object field was averaged over N = 128 frame sequences, and reported in the first column of Fig. 1. The image is not visible for optical power transmission factors lower or equal to 10 -3 (i.e. O.D. = 3), which corresponds to 19.7 photo-electrons per pixel recorded on average during 128 frames. Then, the recorded optical fluctuations were projected on the expected oscillation to retrieve the modulated component to improve the detection sensitivity. The modulation depth of the signal was assessed by a fast Fourier transform (FFT) X of the sequence x k at the modulation frequency ω S /4 (5 Hz), calculated on 128 time points. Results are reported in the second column of Fig. 1. The image is not visible for a power transmission factor lower or equal to 10 -5 (i.e. O.D. = 5), which corresponds to 1.97×10 -1 photo-electron per pixel recorded on average during 128 frames. Finally, we performed singular value analysis of temporal fluctuations. In this approach, we embedded the time series X = {x 1 , ..., x N } of the signal at each pixel in a Hankel matrix of dimension K × (N -K + 1), where K is the number of principal components to seek, and N = 128 is the total number of acquired interferograms. The Hankel matrix D, composed of N -K + 1 lag-shifted copies of truncated values of X(t) has the form

D =       x 1 x 2 . . . x N -K+1 x 2 x 3 . . . . . . . . . . . . . . . x N -1 x K x K+1 . . . x N       (2)
We proceeded by diagonalizing the lag-covariance matrix of X(t), C X = DD t * , where D t * is the adjoint matrix (Hermitian conjugate) of D. The K eigenvalues λ k , k = 1, ..., K of C X , numbered by decreasing order of magnitude, were calculated by SVD [33] of C X . The length of the analysis window K was chosen to be an integer number of expected periods of the modulated signal [34] K = p ω S /∆ω. We used a window length of p = 5 modulation periods (K = 20). Images were rendered by calculating the amplitude of the second eigenvalue at each pixel. They are displayed in Fig. 1 in the third column, as a function of the optical density in the object channel. With this approach, the image is visible down to an optical power transmission factor of 10 -6 (i.e. O.D. = 6), for which the signal wave intensity accounted for a total of 1.97 × 10 -2 recorded photoelectron per pixel during 128 frames, on average. The algorithm used for the SVD is based on the orthogonal triangular decomposition (QR factorization) [START_REF] Golub | Matrix computations[END_REF]. SVD and FFT were performed by the functions svd() and fft() in Matlab 2012a, run on a CentOS 6 Linux 64-bit operating system. The response curve of singular value demodulation was assessed by plotting the quantity 10 log 10 ( S / S 0 ) (in dB) against the value of the optical density in the object channel (Fig. 2). The braces indicate spatial averaging of S, and S 0 within outside the reconstructed image, respectively. The expected ideal detection behavior is plotted as a continuous black line (-10 dB of optical power per unit of optical density). The demodulation by SVD is qualitative [START_REF] Broomhead | [END_REF]; yet it enables to retrieve images at much lower signal levels than FFT demodulation. The singular value demodulation procedure is illustrated by the spectrograms of four signals, reported in Fig. 3 : (a) the complex wave oscillating at ∆ω/(2π) = 5 Hz, which be expected in perfect modulation conditions, in the absence of random phase fluctuation (Eq. 1); (b) the actual signal X(t) from a pixel at O.D. = 2, which exhibits a DC contribution and jitter -deviation from the modulation frequency; and the signal reconstructed from the first (c) and second (d) eigenvectors and eigenvalues of the SVD of D. In this example, the DC term is found in the first component, and the modulated component at 5 Hz undergoing a phase drift is retrieved in the second component of the SVD.

In conclusion, we used heuristically a singular value decomposition procedure to retrieve time-modulated optical signals in holographic interferometry. In the reported experiment, over long acquisition times, singular value demodulation of phase-shifted interferograms enabled qualitative image visualization at much lower signal levels than demodulation by discrete Fourier transformation. This method will be investigated for signal retrieval in strong phase fluctuations conditions, phase microscopy, and singular spectrum analysis of pseudoperiodic modulations. This work was supported by Agence Nationale de la Recherche (ANR-09-JCJC-0113, ANR-11-EMMA-046), Fondation Pierre-Gilles de Gennes (FPGG014), région

5 FFTFIG. 1 :

 51 FIG. 1: Amplitude holographic images from 4-phase modulated interferograms acquired over 128 frames. No time demodulation (first column). Demodulation by Fourier transform (second column). Demodulation by singular value decomposition (third column).

FIG. 2 :

 2 FIG. 2: Signal magnitude versus optical density (O.D.) in the object channel from a sequence of N = 128 frames of 4-phase interferograms. No time demodulation (circles). Demodulation by discrete Fourier transform (squares). Demodulation by singular value decomposition (triangles).

FIG. 3 :

 3 FIG. 3: Spectrograms of a complex wave oscillating at 5 Hz (a), a hologram's pixel signal X(t) (b), reconstructed signals from the first (c) and second (d) elements of the SVD of D.

PSL*), and European Research Council (ERC Synergy HELMHOLTZ).


Jeffrey Bencteux

Pierre Pagnoux

Thomas Kostas

Sam Bayat

Michael Atlan

Holographic laser Doppler imaging of pulsatile blood flow

We report on wide-field imaging of pulsatile motion induced by blood flow using heterodyne holographic interferometry on the thumb of a healthy volunteer, in real-time. Optical Doppler images were measured with green laser light by a frequency-shifted Mach-Zehnder interferometer in off-axis configuration. The recorded optical signal was linked to local instantaneous out-of-plane motion of the skin at velocities of a few hundreds of microns per second, and compared to blood pulse monitored by plethysmoraphy during an occlusion-reperfusion experiment.

INTRODUCTION

One of the most commonly displayed clinical waveforms is the finger plethysmogram.

It allows the non-invasive measurement of pulse wave amplitude, which reflects the changes in finger blood flow. It is usually recorded by a pressure sensor [START_REF] William | Review of medical physiology[END_REF] or a pulse oximeter which illuminates the skin and measures changes in optical absorption [START_REF] Ab Hertzman | Observations on the finger volume pulse recorded photoelectrically[END_REF][START_REF] Allen | Photoplethysmography and its application in clinical physiological measurement[END_REF][START_REF] Aymen | Photoplethysmography[END_REF]. Optical plethysmography is widely available, its feasibility has even been demonstrated using mobile phones [START_REF] Jonathan | Investigating a smartphone imaging unit for photoplethysmography[END_REF][START_REF] Scully | Physiological parameter monitoring from optical recordings with a mobile phone[END_REF]. However, it does not provide a spatially-resolved image. This limitation can be overcome by signal processing of regular video recordings in ambient light for time-dependent spatial and color fluctuations enhancement [START_REF] Takano | Heart rate measurement based on a time-lapse image[END_REF][START_REF] Scalise | Heart rate measurement in neonatal patients using a webcamera[END_REF][START_REF] Wu | Eulerian video magnification for revealing subtle changes in the world[END_REF][START_REF] Balakrishnan | Detecting pulse from head motions in video[END_REF].

Other imaging approaches are designed to harness optical phase fluctuations of coherent laser light from the motion of body walls, either by single-point scanning detection [START_REF] Essex | A laser doppler scanner for imaging blood flow in skin[END_REF], direct image detection of temporal [START_REF] Draijer | Twente optical perfusion camera: system overview and performance for video rate laser doppler perfusion imaging[END_REF][START_REF] Leutenegger | Real-time full field laser doppler imaging[END_REF][START_REF] Kamshilin | Variability of microcirculation detected by blood pulsation imaging[END_REF] and spatial speckle contrast [START_REF] David | Laser speckle con-trast imaging in biomedical optics[END_REF][START_REF] Humeau-Heurtier | Skin perfusion evaluation between laser speckle contrast imaging and laser doppler flowmetry[END_REF], or holographic interferometry [START_REF] Mathieu Leclercq | Evaluation of surface acoustic waves on the human skin using quasitime-averaged digital fresnel holograms[END_REF][START_REF] Dobrev | Optimization of a lensless digital holographic otoscope system for transient measurements of the human tympanic membrane[END_REF].

In this letter, we report on narrow band laser Doppler imaging of superficial pulsatile motion of the thumb by heterodyne holographic interferometry with a camera refreshed at 60 Hz. A transient blood flow interruption experiment in a healthy volunteer is performed and compared to blood pulse monitoring. A major advantage of this system is that no physical contact with the studied tissue surface area is required. This could allow assessment of non-intact skin or mucous membranes or even the intraoperative study of tissues within the surgical field. Moreover, the lack of surface contact reduces the risk of infection transmission which is a true concern with reusable contact-requiring biomedical equipment [START_REF] Mc Wilkins | Residual bacterial contamination on reusable pulse oximetry sensors[END_REF], also avoiding the extra cost of disposable contact probes.

EXPERIMENTAL SETUP

The experimental laser Doppler imaging scheme designed for this study is sketched in Fig. 1. The apparatus consists of a fibered Mach-Zehnder optical interferometer for off-axis [START_REF] Leith | Reconstructed wavefronts and communication theory[END_REF] and frequency-shifting [START_REF] Aleksoff | Time average holography extended[END_REF][START_REF] Macovski | Efficient holography using temporal modulation[END_REF] holographic interferometry. The main optical radiation field is provided by a 150 mW, single-mode, fibered diode-pumped solid-state green laser (Cobolt Samba-TFB-150) at wavelength λ = 532 nm, and optical frequency ω L /(2π) = 5.6 × 10 14 Hz. The thumb is illuminated with about ∼ 10 mW of continuous optical power, over ∼ 40 mm × 40 mm. In the reference channel, an optical local oscillator (LO) is formed by optical frequency shifting. Two acousto-optic modulators (Bragg cells, AA Opto Electronic), driven with phase-locked signals at ω C and ω C + ∆ω, are used to shift the optical frequency of the laser beam from ω L to ω L + ∆ω. The carrier frequency ω C /(2π) is set to 200 MHz, at the peak response of the fibered acousto-optic modulators. The backscattered optical field E is mixed with the LO field E LO with a non-polarizing beam splitter cube. A Ximea MQ042MG-CM camera records interference patterns at a frame rate of τ -1 S = ω S /(2π) = 60 Hz. Each raw interferogram of 2048 × 2048 pixels, digitally acquired at time t is noted

I(t) = |E(t) + E LO (t)| 2 .
A small angular tilt ∼ 1 • between E and E LO ensures off-axis recording. The sensor is set ∼ 50 cm away from the object plane. A convergent lens of 50 mm focal length is present in the object channel, in order to widen the lateral field of view of the holographic detection to ∼ 50 mm. The backscattered laser optical field

E(t) = E exp [iω L t + iφ(t)
] undergoes a phase variation φ(t), as a consequence of out-of-plane motion of the illuminated tissue. It is mixed with the LO field from the reference channel

E LO (t) = E LO exp [iω L t + i∆ωt], that is tuned to a close-by intermediate frequency ω L + ∆ω.
The quantities E and E LO are complex constants and i is the imaginary unit. The magnitude of a given optical Doppler component is retrieved by frequency down conversion within the camera's temporal bandwidth, ensured by non-linear detection of the field E by the array of square-law sensors of the camera, that respond linearly with the optical irradiance

I(t) = |E(t) + E LO (t)| 2 .
The squared magnitude of the total field received, E + E LO , has cross-terms oscillating at the difference frequency of the fields E and E LO

I(t) = |E| 2 + |E LO | 2 + H(t) + H * (t) (1)
where

H(t) = EE * LO exp (iφ -i∆ωt)
is the heterodyne interferometric contribution, and * denotes the complex conjugate. This equation describes the temporal fluctuation of the recorded irradiance at a given pixel.

SIGNAL PROCESSING

Holographic image rendering, or spatial demodulation, is then performed onto each recorded interferogram I with a discrete Fresnel transform [START_REF] Goodman | Digital image formation from electronically detected holograms[END_REF]. In off-axis recording configuration, The Fresnel transform separates spatially the four interferometric terms of the right member of Eq. 1 [START_REF] Leith | Reconstructed wavefronts and communication theory[END_REF][START_REF] Cuche | Spatial filtering for zero-order and twin-image elimination in digital off-axis holography[END_REF]. After spatial demodulation of each interferogram, the heterodyne signal H(t) appears in the offaxis region of the hologram, processed as follows : First, the squared amplitude of the difference of two consecutive off-axis holograms is formed in order to cancel very low frequency noise contributions Then, to cancel laser intensity fluctuations, a normalization factor . Hence the processed Doppler signal S 2 / S 2 0 is a heterodyne measurement of the optical power (or irradiance) |E| 2 , which does not depend on the LO power [START_REF] Magnain | Holographic laser doppler imaging of microvascular blood flow[END_REF].

S 2 = |H(t) -H(t -τ S )| 2 .
S 2 0 = |H 0 (t) -H 0 (t -τ S )|

NARROWBAND APPARATUS RESPONSE

Two-phase demodulation of interferograms recorded with a sampling frequency approximately equal to the reciprocal of the exposure time (τ -1 S ≃ τ -1 E ) results in a narrowband detection with an apparatus response plotted in Fig. 2. This response filters-off optical contributions which are not shifted in frequency, i.e. staticallyscattered light. For the backscattered field's irradiance |E| 2 , this response is [START_REF] Michael Atlan | Parallel heterodyne detection of dynamic light-scattering spectra from gold nanoparticles diffusing in viscous fluids[END_REF][START_REF] Verrier | Laser doppler holographic microscopy in transmission: application to fish embryo imaging[END_REF]

B(ω) = 1 ω 2 τ 2 E sin 2 ωτ S 2 sin 2 ωτ E 2 ( 2
)
where ω is the apparent frequency recorded by the camera, as a result of heterodyne beating. Let's assume that at the short time scale of the sampling process t < τ S = 2π/ω S , the local instant velocity is approximately constant. v(t) ≈ v(t -τ S ) ≈ v. In other words, its magnitude v is identified to the time-averaged value of the instant velocity during τ S , so we can simply write the local optical phase variation due to the transverse displacement of the skin as φ(t) = 2kvt = ωt, when the illumination and the scattered wave vectors are perpendicular to the skin's surface. The optical wave number is k = 2π/λ ≃ 1.2 × 10 7 rad.s -1 , and ω = 2kv. Now let's consider temporal variations of the instant velocity of norm v(t) at long time scales t > τ S . The light reflected by the skin undergoes a local time-dependent Doppler shift

ω(t) = 2kv(t) (3)
To assess a given instant velocity v 0 , the LO is detuned by ∆ω = 2kv 0 . According to Eq. 3 and Eq. 2, it results in the detection of Doppler-shifted light with a velocitydependent efficiency B (2k(v -v 0 )). The range of probed velocities v within the two first lobes of the apparatus response B is

[v -1 , v +1 ] = v 0 - ω S 2k , v 0 + ω S 2k (4)
EXPERIMENT An experiment on the fingers of healthy volunteers undergoing arterial occlusion provoked with a pressure cuff was conducted. The pressure cuff was placed around the right arm. The cuff bladder pressure was measured using a transducer (MLT0670, ADInstruments, Oxford, UK). The signal was amplified, digitized at 200 Hz using an analog/digital converter (Powerlab; ADI Instruments), and recorded on a computer. The arterial pulse wave was monitored using a piezo-electric pulse transducer (TN1012/ST, ADInstruments) placed on the index of the imaged hand. The signal was recorded at 200 Hz. The thumb was illuminated by the green laser beam. A comfortable support for the forearm was used avoid stray motion. A preliminary experiment was performed to monitor the Doppler response at three different detuning frequencies ∆ω/(2π) : 60 Hz, 600 Hz, and 1020 Hz; choosing a detuning frequency which is an integer number of times the sampling frequency is not mandatory, but is has the advantage of canceling efficiently staticallybackscattered light, according to the apparatus response (Fig. 2). These frequencies correspond to probed velocities of 16 µm.s -1 , 160 µm.s -1 , and 271 µm.s -1 , respectively. Fig. 3 shows plots over time of pulse waveforms (plethysmograms) and holographic Doppler signal 10 log 10 S 2 / S 2 0 (in dB) averaged in a region of interest, placed on the central part of the image of the thumb. At very low frequency detunings ( 100 Hz), motion artifacts prevented the measurement of pulsatile motion. When the frequency was set to ∆ω/(2π) = 1020 Hz, which corresponded to a measurement range of instant velocities [v -1 , v +1 ] = [255 µm.s -1 , 287 µm.s -1 ], centered at v 0 = ∆ω/(2k) = 271 µm.s -1 (Eq. 4), the Doppler signal appeared to be less prone to motion artifacts. The

CONCLUSION

In conclusion, we demonstrated wide-field imaging and monitoring of pulsatile motion of the thumb of a healthy volunteer. Narrowband optical Doppler video imaging in real-time was performed with a off-axis and frequencyshifting holographic interferometer. Numerical image rendering was performed by discrete Fresnel transformation and two-phase temporal demodulation at a 60 Hz framerate. The measured contrast was linked to the instant velocity of out-of-plane motion, of the order of a few hundreds of microns per second. Robust non-contact motion monitoring was achieved. This was illustrated by an occlusion-reperfusion experiment, in which the proposed detection scheme was compared to plethysmography.

FIG. 1 :

 1 FIG.1: Sketch of the fibered Mach-Zehnder heterodyne holographic interferometer. The main laser beam is split into two channels. In the object channel, the optical field E is backscattered by the skin. In the reference channel, the optical field ELO is frequency-shifted by two Bragg cells by ∆ω. A standard camera records interferograms of the diffracted fields E and ELO.

FIG. 2 :

 2 FIG.2: Apparatus response (Eq. 2), calculated for τS = τE, versus apparent (beating) frequency index n = ω/ωS. The ticks of the horizontal axis correspond to the zeros of the response at integer values of the index n of the beating frequency nωS and the probed velocity vn = (∆ω + nωS)/(2k).

FIG. 3 :FIG. 4 :

 34 FIG.3: Doppler signal (a,c,e), averaged in the central part of the image of the thumb (Fig.4), and concurrent blood volume signal (b,d,f), versus time. Detuning frequencies : ∆ω/(2π) = 60 Hz (a), ∆ω/(2π) = 600 Hz (c), ∆ω/(2π) = 1020 Hz (e). The synchronization accuracy of holographic and standard blood volume measurements is of the order of ±0.5 s. a.u.: arbitrary units.

FIG. 5 :

 5 FIG. 5: Time traces of indicators monitored during a hypoperfusion experiment. Cuff pressure (a), plethysmogram [raw data (b), 5 s time-average of the magnitude (c)], holographic laser Doppler signal recorded at 1020 Hz and averaged spatially in the nail area [raw data (d), 5 s time-average (e)]. A time-lapse sequence of the composite Doppler image of the thumb is reported in this video.

  2 is formed. The contribution H 0 is measured in a region of interest of the reconstructed hologram where the terms of Eq. 1 are not present[START_REF] Atlan | Laser doppler imaging, revisited[END_REF], and the brackets account for spatial averaging. In high heterodyne gain regime, i.e. when |E| 2 ≪ |E LO | 2 , the quantity S 2 0 is dominated by shot-noise of the LO[START_REF] Gross | Digital holography with ultimate sensitivity[END_REF][START_REF] Lesaffre | Noise and signal scaling factors in digital holography in weak illumination: relationship with shot noise[END_REF], which scales up linearly with |E LO |

2 . The averaged signal S 2 scales up linearly with |E| 2 |E LO | 2

  5 s-averaged pulse wave magnitude, Doppler signal, and 5 s-averaged Doppler signal. Blood flow to the forearm was occluded by increasing the cuff pressure to ∼ 200 mmHg during a ∼ 90 s time interval. During occlusion, both recorded signals dropped. Short spikes due to motion artifacts were observed on the Doppler signal; they were more pronounced at lower Doppler frequency. On releasing the cuff pressure, a transient increase of both the pulse volume and the Doppler signal is observed as expected, due to post-occlusive hyperemia. A time-lapse sequence of the composite Doppler image of the thumb

	during the occlusion-reperfusion experiment is reported
	in this video. A limitation of holographic Doppler imag-
	ing of pulsatile blood flow with visible laser light is its
	sensitivity to low velocities, which lets motion artifacts
	prevent the measurement of pulsatile motion, and re-
	quires stabilization of the monitored tissue. This issue
	might be alleviated with infrared light.

Acknowledgments

We gratefully acknowledge the contributions of Anna Kieblesz, Renaud Boistel, Aurore Bonnin, and Claude Boccara, and financial support from Agence Nationale de la Recherche (ANR-09-JCJC-0113, ANR-11-EMMA-046), Fondation Pierre-Gilles de Gennes (FPGG014), région Ile-de-France (C'Nano, AIMA), the Investments for the Future program (LabEx WIFI: ANR-10-LABX-24, ANR-10-IDEX-0001-02 PSL*), and European Research Council (ERC Synergy HELMHOLTZ).


Som Sharma

Prashant Kumar
email: somkumar@prl.res.in

Chintan Jethva

Rajesh Vaishnav

Hassan Bencherif

Investigations of the middle atmospheric thermal structure and oscillations over sub-tropical regions in the Northern and Southern Hemispheres

Keywords: Atmospheric dynamics, Middle Atmosphere, SABER, Atmospheric models

In upper mesosphere, strength of annual oscillations is substantial in NH, while semi-annual oscillations are stronger in SH. Wavelet analyses found that annual oscillations are significant in NH near mesopause, while semi-annual oscillations are strengthening in SH.

Introduction

Northern Hemisphere (NH) and Southern Hemisphere (SH) have varying and diverse geophysical processes due to differences in land cover, ocean extents, radiative fluxes, etc. NH has about 90 % of the population while SH has large oceanic region. Due to lesser human existence in the SH, studies of the Earth's atmosphere are relatively less in general, and in particular, middle atmospheric studies are sparser as it is rather difficult to probe middle atmosphere from ground based instruments. Thermal structure is one of the efficient approach to understand inter-hemispheric coupling of the middle atmosphere. [START_REF] Kornich | A simple model for the interhemispheric coupling of the middle atmosphere circulation[END_REF] found that anomalous stratospheric and mesospheric features are due to residual circulation pattern. To enhance our scientific understanding of the middle atmosphere, it is crucial to compare the NH and SH processes using similar kind of observations from various observing platforms and numerical models. [START_REF] Hauchecorne | Climatology and trends of the middle atmospheric temperature (33-87 km) as seen by Rayleigh lidar over the south of France[END_REF] were the first to establish middle atmospheric temperature climatology using Rayleigh Lidar observations from two stations in France. Later, [START_REF] Gobbi | Lidar observations of middle atmosphere temperature variability[END_REF] presented a mid-latitude temperature variability over Frascati (42°N, 13.6° E). Temperature climatology of the middle atmosphere (from ~30 to Abstract The temperature retrieved from the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) onboard Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) satellite during January 2002 to September 2015 are used in this study to delineate the differences of middle atmospheric thermal structure in the Northern Hemisphere (NH) and Southern Hemisphere (SH). Two stations namely Mt. Abu (24.59°N, 72.70°E) in NH and Reunion Island (21.11°S,55.53°E) in SH are chosen over sub-tropical regions. Temperature climatology from SABER observations suggests that stratopause is warmer, and upper mesosphere is cooler in NH as compared to SH. Three atmospheric models are used to understand the monthly thermal structure differences for different altitudes. Moreover, semi-annual, annual and quasi-biennial oscillations are studied using Lomb Scargle Periodogram and Wavelet transform techniques. Over NH, summer and winter season are warmer (~4 K) and cooler (~3 K) respectively in stratosphere as compared to SH. It is important to note here that Mt. Abu temperature is warmer (~9 K) than Reunion Island in winter but in summer season Mt. Abu temperature is cooler in upper mesosphere and above mesosphere NH shows warming. Results show that annual oscillations are dominated in both hemisphere as compared to semi-annual and quasi-biennial oscillations. ~90 km) have been studied by several groups (e.g., [START_REF] Barnett | Middle atmosphere reference model derived from satellite data[END_REF][START_REF] Chanin | Contribution to the CIRA model from ground based lidar[END_REF][START_REF] Jenkins | Upper stratospheric and mesospheric temperatures derived from lidar observations at Aberystwyth[END_REF][START_REF] Hauchecorne | Climatology and trends of the middle atmospheric temperature (33-87 km) as seen by Rayleigh lidar over the south of France[END_REF][START_REF] Clancy | Temperature minima in the average thermal structure of the middle atmosphere (70-80 km) from analysis of 40-to 92-km SME global temperature profiles[END_REF][START_REF] Whiteway | Rayleigh lidar observations of thermal structure and gravity wave activity in the high arctic during a stratospheric warming[END_REF][START_REF] Leblanc | Temperature climatology of the middle atmosphere from long-term lidar measurements at middle and low latitudes[END_REF][START_REF] Bencherif | Observation and first validation of stratospheric temperature profiles obtained by a Rayleigh-Mie LIDAR over Durban, South Africa[END_REF][START_REF] Gardner | First Lidar observations of middle atmosphere temperatures, Fe densities, and polar mesospheric clouds over the North and South Poles[END_REF][START_REF] Nee | Middle atmospheric temperature structure over two tropical locations, Chung-Li (251 N, 1211E) and Gadanki (13.51 N,79.21E[END_REF][START_REF] Sivakumar | Lidar measurements of stratosphere-mesosphere thermal structure at a low latitude: comparison with satellite data and models[END_REF][START_REF] Randel | The SPARC intercomparison of middle atmospheric climatologies[END_REF][START_REF] Argall | A comparison of Rayleigh and sodium lidar temperature climatologies[END_REF][START_REF] Gerding | Seasonal variation of nocturnal temperatures between 1 and 105 km altitude at 541 N observed by lidar[END_REF][START_REF] Li | Interannual variations of middle atmospheric temperature as measured by the JPL lidar at Mauna Loa Observatory[END_REF][START_REF] Batista | Tidal associated temperature disturbances observed at the middle atmosphere (30-65 km) by a Rayleigh LIDAR at 23o[END_REF][START_REF] Batista | A 14-year monthly climatology and trend in the 35-65 km altitude range from Rayleigh LIDAR temperature measurements at a low latitude station[END_REF][START_REF] Kumar | Mean thermal structure of the low-latitude middle atmosphere studied using Gadanki Rayleigh lidar, Rocket, and SABER/TIMED observations[END_REF][START_REF] Dou | Seasonal oscillations of middle atmosphere temperature observed by Rayleigh LIDARs and their comparisons with TIMED/SABER observations[END_REF][START_REF] Sivakumar | Rayleigh Lidar and Satelite (HALOE, SABER, CHAMP and COSMIC) measurements of stratopshere-mesosphere temperature over southern sub-tropical site, Reunion (20.81S, 55.51E): climatology and comparison study[END_REF][START_REF] Sharma | A comprehensive study on middle atmospheric thermal structure over a low and near mid-latitude stations[END_REF][START_REF] Sharma | Middle atmospheric thermal structure over sub-tropical and tropical Indian locations using Rayleigh lidar[END_REF][START_REF] Sharma | Study of thermal structure differences from coordinated lidar observations over Mt. Abu (24.5° N, 72[END_REF]. Thermodynamic profiles from troposphere to mesosphere were studied by various groups using Lidars (e.g., [START_REF] Hauchecorne | Lidar monitoring of the temperature in the middle and lower atmosphere[END_REF]Gross et al. 1997 and reference therein). [START_REF] Namboothiri | Rayleigh lidar observations of temperature over Tsukuba, winter thermal structure and comparison studies[END_REF] compared satellite retrieved temperature profiles and CIRA-86 (COSPAR International Reference Atmosphere) model with Lidar and Rocket measured profiles and found that the Lidar temperatures are agreeing well with rocket and satellite borne measurements. Ground based Lidar measured temperatures were also used for the validation of several remote sensing instruments e.g. Microwave Limb Sounder (MLS), HALOE (Halogen Occultation Experiment) onboard UARS (e.g., [START_REF] Remsberg | An assessment of the Quality of HALOE temperature profiles in the mesosphere with Rayleigh Backscatter Lidar and inflatable falling sphere measurements[END_REF]Cooper 2004, Sica et al. 2008 andreferences therein). Over the Indian tropical latitudes in NH, thermal structure was studied initially from rocket borne observations [START_REF] Sasi | A model equatorial atmosphere over the Indian zone from 0 to 80 km[END_REF][START_REF] Lal | Equatorial stratospheric and meso-spheric structural variations during the years 1971-74[END_REF][START_REF] Sasi | A reference atmosphere for the Indian equatorial zone[END_REF][START_REF] Mohankumar | Temperature variability over the tropical middle atmosphere[END_REF]) and later using ground based Lidar measurements (e.g., [START_REF] Parameswaran | Altitude profiles of temperature from 4-80 km over the tropics from MST radar and lidar[END_REF][START_REF] Sivakumar | Lidar measurements of stratosphere-mesosphere thermal structure at a low latitude: comparison with satellite data and models[END_REF][START_REF] Chandra | Rayleigh Lidar studies of thermal structure over Mt Abu[END_REF][START_REF] Sharma | A comprehensive study on middle atmospheric thermal structure over a low and near mid-latitude stations[END_REF][START_REF] Sharma | Middle atmospheric thermal structure over sub-tropical and tropical Indian locations using Rayleigh lidar[END_REF][START_REF] Sharma | Study of thermal structure differences from coordinated lidar observations over Mt. Abu (24.5° N, 72[END_REF]. A study using 14 years of Lidar data on the seasonal thermal amplitude were reported by [START_REF] Batista | A 14-year monthly climatology and trend in the 35-65 km altitude range from Rayleigh LIDAR temperature measurements at a low latitude station[END_REF] at Sao Jose dos Campos, Brazil (23°S, 46°W) in the SH and found significant differences in the temperatures simulated from the atmospheric model. Another Lidar based study of middle atmospheric thermal structure in the SH at Reunion Island (21.11°S, 55.53°E) was presented by [START_REF] Sivakumar | Rayleigh Lidar and Satelite (HALOE, SABER, CHAMP and COSMIC) measurements of stratopshere-mesosphere temperature over southern sub-tropical site, Reunion (20.81S, 55.51E): climatology and comparison study[END_REF]. They reported that the Lidar observed profiles a re i n q ualitative a greement w ith t he s atellite o bservations. In addition, few studies are available in SH covering low latitudes and sub-tropics (e.g., [START_REF] Bencherif | Observation and first validation of stratospheric temperature profiles obtained by a Rayleigh-Mie LIDAR over Durban, South Africa[END_REF][START_REF] Morel | Evidence of tidal perturbations in the middle atmosphere over Southern Tropics as deduced from LIDAR data analysis[END_REF][START_REF] Batista | Tidal associated temperature disturbances observed at the middle atmosphere (30-65 km) by a Rayleigh LIDAR at 23o[END_REF][START_REF] Batista | A 14-year monthly climatology and trend in the 35-65 km altitude range from Rayleigh LIDAR temperature measurements at a low latitude station[END_REF]. Vertical profiles of temperature are also available from HALOE onboard UARS (1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005), COSMIC (Constellation Observing System for Meteorology, Ionosphere, and Climate) and other GPSRO (Global Positioning System Radio Occultation) mission, hyperspectral infrared sounder on board AQUA, and SABER (Sounding of the Atmosphere using Broadband Emission Radiometry) onboard TIMED (Thermosphere Ionosphere Mesosphere Energetics and Dynamics). Simultaneous observations in the stratosphere and mesosphere from these instruments have further provided a platform to study the coupling between different regions and this further lead to improved skill of model predictions.

Gravity waves (GW) and planetary waves (PW) characteristics are highly varying regionally and seasonally over the globe, and have significant differences in the NH and SH [START_REF] Fritts | Spectral estimates of gravity wave energy and momentum fluxes. Part I: energy dissipation, acceleration, and constraints[END_REF][START_REF] Fritts | Gravity wave dynamics and effects in the middle atmosphere[END_REF]. Waves of non-orographic origin modulate winds at mesospheric heights, and even affect total stratosphere in the tropical and sub-tropical regions, and are one of the major drivers of the Quasi Biennial Oscillations (QBO) and Semi-Annual Oscillations (SAO) (e.g., [START_REF] Alexander | Nonstationary gravity wave forcing of the stratospheric zonal mean wind[END_REF][START_REF] Baldwin | The quasi-biennial oscillation[END_REF][START_REF] Giorgetta | Forcing of the quasibiennial oscillation from a broad spectrum of atmospheric waves[END_REF][START_REF] Pulido | The seasonal cycle of gravity wave drag in the middle atmosphere[END_REF]Sato et al. 2009 and reference therein). The SAO are also playing a vital role in deciding overall dynamics of the Earth's atmosphere [START_REF] Garcia | Climatology of the semiannual oscillation of the tropical middle atmosphere[END_REF]. Features of stratospheric stationary planetary waves (SPW) in the NH and SH was presented by various authors (e.g., [START_REF] Hirota | Upper stratospheric circulations in the two hemispheres observed by satellites[END_REF][START_REF] Hartmann | Observations of wave-mean flow interaction in the Southern Hemisphere[END_REF][START_REF] Mechoso | Climatology and interannual variability of wave, mean-flow interaction in the Southern Hemisphere[END_REF][START_REF] Shiotani | Planetary wave-mean flow interaction in the stratosphere: a comparison between Northern and Southern Hemispheres[END_REF][START_REF] Andrews | Some comparisons between the middle atmosphere dynamics of the southern and northern hemispheres[END_REF]. [START_REF] Forbes | Climatological features of mesosphere and lower thermosphere stationary planetary waves within ± 40 latitude[END_REF] reported PW structures and their seasonal variability in the altitude range of 15-100 km using data from High Resolution Doppler Imager (HRDI) and Wind Imaging Interferometer (WINDII) onboard Upper Atmosphere Research Satellite (UARS). [START_REF] Gray | Solar influences on climate[END_REF] reported impact of QBO on middle atmospheric temperature and dynamics. During winter in the NH, easterly phase of QBO is responsible for poleward as well as downward propagating easterly wind anomalies [START_REF] Matthes | The importance of time-varying forcing for QBO modulation of the atmospheric 11 year solar cycle signal[END_REF].

Latitudinal coupling is another very pertinent aspect for the study of atmospheric system in whole. Tropical and sub-tropical regions are very much prone to be influenced by different hemispheres as being close to the equator. Stronger equatorial processes in either of the hemispheres modulates each other. Therefore, it is very much needed to explore different atmospheric regions of the NH and SH to quantify the similarities, differences and anomalies in the tropical and sub-tropical regions of the different hemispheres. Present study is focussed on the middle atmospheric thermal structures and characteristics of SAO, AO (Annual Oscillations) and QBO over two regions, Mt. Abu in the NH, and Reunion Island almost at similar latitude in the SH, using SABER onboard TIMED satellite during January 2002 to September 2015. Similarities and differences in thermal fields and oscillations have been extensively analysed, and findings are discussed using Lomb-Scargle Periodogram, and to further delineate the presence of SAO, AO and QBO using wavelet techniques.

Instrumentation and data

In this paper, we have used observations from SABER onboard TIMED satellite from January 2002 to September 2015 over Mt. Abu, a subtropical location in the NH, and over Reunion Island which is located in the sub-tropics of the SH (Fig. 1). Radiometric technique have been exploited effectively to retrieve vertical temperature profiles in SABER. It measures radiations from 1.27 to 17 µm covering ten spectral channels in limb direction. The temperature in the stratosphere-mesosphere-lower thermosphere is retrieved from atmospheric CO 2 emission at 15 µm [START_REF] Mertens | Retrieval of mesospheric and lower thermospheric kinetic temperature from measurements of CO2 15-mm Earth limb emission under non-LTE conditions[END_REF][START_REF] Mertens | Kinetic temperature and carbon dioxide from broadband infrared limb emission measurements taken from the TIMED/SABER instrument[END_REF]. The SABER on-board TIMED satellite covers 15 orbits in a day at an altitude of about 625 km having an orbital inclination of 74.1° to the equator. [START_REF] Mertens | Retrieval of mesospheric and lower thermospheric kinetic temperature from measurements of CO2 15-mm Earth limb emission under non-LTE conditions[END_REF] and [START_REF] Remsberg | Assessment of the quality of the Version 1.07 temperature-versus-pressure profiles of the middle atmosphere from TIMED/SABER[END_REF] presented an extensive study on the observational and retrieval errors in the SABER observed temperatures. [START_REF] Mertens | Retrieval of mesospheric and lower thermospheric kinetic temperature from measurements of CO2 15-mm Earth limb emission under non-LTE conditions[END_REF] reported uncertainty in the retrieval and found that it is about 1.4 K at 80 km and 22.5 K at 110 km. The SABER measurements have been used in the 5° × 5° latitude-longitude grid, centred over Mt. Abu and Reunion. 3 Results and discussion

Temporal variations

The aim of this study is to understand the characteristics of sub-tropical middle atmosphere in NH and SH. To understand the behaviour of stratosphere-mesosphere thermal structure in two different hemispheres, SABER onboard TIMED satellite observations are used in this study from 20 to 100 km vertically during January 2002 to September 2015. Temporal variation of SABER retrieved monthly mean temperature during study period for Reunion station is shown in Fig. 2a, and the moving annual average temperature difference between Mt. Abu and Reunion temperature observations is depicted in Fig. 2b. As compared to Mt. Abu, stratopause temperature (at ~47 km) is relatively less (~2 K) in Reunion Island, which can be easily detected from Fig. 2b. It is interesting to note that Mt. Abu temperatures are higher than Reunion temperatures above 70-80 km. These differences reach up to larger than 5 K. In general, mean climatological behaviour of upper mesosphere shows that NH has higher temperature as compared to SH. [START_REF] Kang | Croll revisited: why is the northern hemisphere warmer than the southern hemisphere?[END_REF] used surface temperature data and re-investigated Croll's 140-year-old theory [START_REF] Croll | V.-The Boulder-clay of Caithness a Product of Landice[END_REF] that NH is warmer than SH due to northward crossequatorial ocean heat transport. This may be due to a transient response of greenhouse gas forcing because the NH has larger land regions which warm faster than oceanic region [START_REF] Philander | Why the ITCZ is mostly north of the equator[END_REF][START_REF] Kang | Croll revisited: why is the northern hemisphere warmer than the southern hemisphere?[END_REF]). The different characteristics of lower tropospheric temperature affect the Gravity waves, which influences subsequently to middle atmosphere. [START_REF] Fritts | Gravity waves in the middle atmosphere of the southern hemisphere[END_REF] studied that both hemispheres reveal similar Gravity waves characteristics in lower and middle atmosphere mainly in case of scales, growth of amplitude with height. However, significant differences were found by authors in the wave spectra and their influences in different hemispheres. Gravity waves transport momentum from lower atmosphere to upward direction (e.g., [START_REF] Fritts | Gravity wave dynamics and effects in the middle atmosphere[END_REF][START_REF] Sato | Gravity wave characteristics in the Southern Hemisphere revealed by a high-resolution middle-atmosphere general circulation model[END_REF]. [START_REF] Becker | High Rossby-wave activity in austral winter 2002: Modulation of the general circulation of the MLT during the MaCWAVE/MIDAS northern summer program[END_REF] found inter-hemispheric coupling of the middle atmospheric circulations in a case study. [START_REF] Kornich | A simple model for the interhemispheric coupling of the middle atmosphere circulation[END_REF] also discussed the inter-hemispheric coupling and essential steps of this coupling were presented in details by [START_REF] Karlsson | Inter-hemispheric mesospheric coupling in a comprehensive middle atmosphere model[END_REF]. In mesosphere at 70-80 km, Fig. 2b shows that temperature is higher in NH against SH. Monthly climatology is prepared using SABER retrieved temperature profiles from 14 years of observations. Figure 3 shows that higher than 250 K temperature is observed at stratopause region over Mt. Abu as well as over Reunion Island. The NH (Fig. 3a) shows maximum warming in April and May months (~265 K), this higher temperature values are not found in Reunion Island (Fig. 3b). Mt. Abu region shows cooler temperature in upper mesosphere during winter season (November to January) over NH. This cold temperature reaches up to 90 km altitude. In contrary to these changes in the NH, the SH also shows cooler temperature in the months of June and July. The difference between Mt. Abu and Reunion temperature (Fig. 3c) shows that higher temperature is found in upper mesosphere except May to September months. Cooler temperature is seen in mesopause over SH during April to October months. Moreover, high temperature is found in lower mesosphere in NH compared to SH during November to February months. In general, stratosphere shows higher temperature at Mt. Abu compared to Reunion Island (Fig. 3c).

Climatology of monthly temperature anomaly, defined as difference between mean monthly temperature (e.g. average values of January for all years) minus mean annual temperature, is prepared using SABER observations. Figure 4a, b shows the temperature anomaly for Mt. Abu and Reunion Island respectively for stratosphere-mesosphere system. Both hemispheres show similar temperature structure in stratosphere, which are higher during February to April and November months and marginally less in the months of August. Somewhat positive anomaly is found in the SH as compared to the NH in the months of December to February (winter period in the NH) within 50-70 km. These anomalies are reverse in nature during August and September months, represents positive value of anomaly in Mt. Abu as compared to Reunion Island. The NH (Fig. 4a) shows coolest temperature at ~85 km in the months of February, and this cooling is shifted at lower altitude from February to September months, which followed by a warmer anomaly trend. Almost similar features are visible in the SH at Reunion Island.

To understand the seasonal variation (e.g. winter, summer season) in both hemisphere, mean temperature during winter and summer for Mt. Abu and Reunion Island and their differences are shown in Fig. 5. Winter and summer season used in this study are December-January-February and June-July-August months respectively in NH. For SH, June-July-August and December-January-February months are selected as winter and summer period respectively. During winter season, stratosphere is cooler over Mt. Abu. At stratopause, both stations show nearly same mean temperature, and in upper mesosphere Mt. Abu is warmer than Reunion Island. In summer season, Mt. Abu shows slightly higher temperature in stratosphere. Above this Fig. 9 A continuous Wavelet transform of the temperature retrieved from SABER satellite representing temporal variations of frequency at every 10 km altitude over Mt. Abu region. A time series of temperature at same altitude during January 2002 to September 2015 is shown in lower panel. Black line in the contour plot shows cone of confidence. Line plot of power and confidence interval at all altitudes are also shown in the right side panel of each contour plot ▸ altitude, Mt. Abu shows cooler temperature within ~67 to 87 km, and further shows more warming as compared to Reunion Island. These results show that the NH is warmer and cooler in summer and winter season respectively as compared to SH in stratosphere, and opposite nature is found in upper mesosphere. This diverse nature may be due to large landmass and high low level warming in the NH.

Figure 5 shows that both the hemispheres have systematic changes in middle atmosphere during two different seasons. This finding motivate us to further investigate the monthly variation of thermal structure. In addition to SABER observations, three different atmospheric models named as CIRA-86, Mass Spectrometer and Incoherent Scatter Radar Extended (MSISE-90) and Naval Research Laboratory Mass Spectrometer and Incoherent Scatter Radar (NRLMSIS-00) are used to evaluate the thermal structure of atmospheric models. Figures 6 and7 show the monthly mean temperature for Mt. Abu and Reunion Island respectively. To calculate monthly temperature, SABER observations and model simulations are used during year 2002 to 2015. Figures 6 and7 show that model's simulated temperature profiles are in good agreement with SABER observations in both hemispheres up to ~30 km. The possible reason may be that various atmospheric sounder in microwave and infrared frequency are available from last few decades which provide temperature and moisture profiles, and very accurate in nature up to ~30 km. These satellite observations are used directly or indirectly in atmospheric models because of that quality of temperature sounding is improved a lot in troposphere and lower stratosphere. At upper stratosphere, slightly higher value of temperature are simulated from atmospheric models as compared to temperature observed from SABER. These differences are maximum in the months of March and April over Mt. Abu region (Fig. 6) and minimum in the months of November. Over Reunion Island, minimum changes are found in the month of June and these changes are higher in March, April and December months. It is interesting to note here that at these vertical levels, all selected atmospheric models show similar kind of warming in the NH, and no significant differences are found in monthly variations. But in case of SH, CIRA-86 model perform better in winter period compared to other two atmospheric models, and worse in summer season at upper stratosphere. In the upper mesosphere around 60 to 80 km, SABER shows minimum temperature for all months over different hemisphere. The differences between atmospheric model's simulations and SABER observations are very high in the months of January, February, and October to December (mainly winter period in NH). These differences are minimum during June to August months for same altitude. In contrary to these results in the NH, the SH (Fig. 7) shows less difference in model simulations and SABER observations at upper mesosphere (~60-80 km) except April, May and September to October months. Overall, we found that somewhat cooler temperature is seen in NH as compared to SH in upper mesosphere. In mesopause region, MSISE-90 and NRLMSIS-00 models are not able to capture temperature accurately for all months over the NH and SH, both models show similar kind of under-estimation at this altitude. But, CIRA-86 model simulated temperature values are close to SABER observations. Slightly more skill is found in the SH as compared to NH temperature at this altitude in the months of January and February. Tables 1 and2 show the SABER observed and model simulated temperature at every 10 km start from 20 to 100 km. Table shows that Reunion Island has minutely higher (~2 K) temperature at 80 km in comparison to Mt. Abu. Maximum error in atmospheric models can be seen at high altitude, which is relatively higher in the NH (Table 1) as compared to the SH (Table 2). In both hemispheres, slightly higher (~4 K) and lesser (~8 K) temperature are simulated from models at 20 and 90 km respectively. Temperature simulated from CIRA-86 is significantly less at 90 km as compared to MSISE-90 and NRLMSIS-00 atmospheric model in both hemispheres. [START_REF] Batista | A 14-year monthly climatology and trend in the 35-65 km altitude range from Rayleigh LIDAR temperature measurements at a low latitude station[END_REF] also studied the seasonal thermal amplitude using 14 years of Lidar measurements and found noticeable differences in the temperatures observed from the MSISE-90 model. Similar kind of comparison are also found in this study when model temperature profiles are compared with SABER observations.

Characteristics of annual, semi-annual and quasi-biennial oscillations

To understand the nature of oscillations in NH and SH, the Lomb Scargle periodogram (LSP;[START_REF] Scargle | Studies in astronomical time series analysis. II-Statistical aspects of spectral analysis of unevenly spaced data[END_REF][START_REF] Lomb | Least-squares frequency analysis of unequally spaced data[END_REF][START_REF] Sivakumar | Rayleigh Lidar and Satelite (HALOE, SABER, CHAMP and COSMIC) measurements of stratopshere-mesosphere temperature over southern sub-tropical site, Reunion (20.81S, 55.51E): climatology and comparison study[END_REF] and Wavelet transform [START_REF] Farge | Wavelet transforms and their applications to turbulence[END_REF][START_REF] Das | Long-term variations in oxygen green line emission over Kiso, Japan, from ground photometric observations using continuous wavelet transform[END_REF] techniques are used for SABER observations during year 2002-2015 in the stratospheremesosphere system. This study will help to distinguish the wave dynamics or wave features of the middle atmospheric region. Previously, [START_REF] Sivakumar | Rayleigh Lidar and Satelite (HALOE, SABER, CHAMP and COSMIC) measurements of stratopshere-mesosphere temperature over southern sub-tropical site, Reunion (20.81S, 55.51E): climatology and comparison study[END_REF] used LSP to study seasonal oscillations from Lidar observations over Reunion Island. The objective of this study is to establish resemblances and alterations over the NH and SH.

Figure 8 shows the different oscillations viz. semi-annual (SAO), annual (AO) and quasi-biennial (QBO) oscillations at every 10 km start from 20 to 90 km for both locations selected in the NH and SH. At 20 km, Mt. Abu as well as Reunion Island show AO, which are stronger over Reunion Island. Interestingly, figure shows that SAO and QBO are exist in NH. SAO and QBO exist in SH but not significant at 95 % confidence interval. Except 50 and 60 km, both locations show AO. These oscillations are stronger in NH as compared to SH at 70-90 km. The QBO are dominant at altitude of 20-40 km over Mt. Abu region, whereas, these oscillations lie at 40 km over the SH (Fig. 8). Less than 6 months' oscillations are also observed in stratopause and lower mesosphere region. Overall, these results suggest that AO are dominated in both hemispheres as compared to SAO and QBO. The strength of AO is almost similar for both hemispheres in stratosphere, but more in upper mesosphere in NH. Contradictorily, SAO are stronger in SH as compared to NH. We also used every 1 km SABER temperature observations for oscillations studies using LSP from 20 to 90 km (figure not shown). It suggests that QBO are stronger in NH at 22 km. These oscillations also exist at 38 km but very less in amplitude for Reunion Island. The AO are not found in the range of 55-63 km in the NH, but the SH exhibits AO at these altitudes. Quarterly oscillations also exist in upper mesosphere in SH, which misses in NH.

Figures 9 and 10 show the wavelet transform at every 10 km from lower stratosphere to mesopause over Mt. Abu and Reunion Island respectively. Time series of SABER temperature observations are shown in lower panel, and upper panel shows corresponding wavelet spectra. The solid black line curve shows cone of influence and right panel shows power of oscillations and red dashed line shows confidence interval. Figure 9 and 10 show that AO are stronger at 20 km in both hemispheres at 95 % confidence interval. Mt. Abu shows SAO which are not very prominent in Reunion Island during selected time period. During year 2009, AO are not seen over Mt. Abu region at 20 km, however AO are significant in the SH for this period. Similar kind of oscillations are seen at 30 km in both the hemisphere, which also confirm by LSP analyses. Wavelet analyses also represent SAO and quarterly oscillations at this altitude which are not matches in the SH and NH. Wavelet spectra at 40 km shows AO and SAO at both selected locations, but QBO are seen significant over Mt. Abu region in the NH. Wavelet spectra very clearly demonstrate that AO are absent at 50 km in NH and SH and mainly dominated by less than annual periodicity. Similar features are also seen at 60 km, while SAO are noteworthy in SH as compared to NH at this altitude. In 70-90 km, Figs. 9 and 10 show that AO are significant at 70 km and these oscillations are not prominent at 90 km. We found a very interesting results for AO at 80 km, which shows that AO are significant mostly over Mt. Abu region but not dominated throughout the period in the SH at this altitude. However, SAO are more strength in southern region as compared to Mt. Abu. Overall, we found that both hemispheres are dominated by AO and SAO and these oscillations are different in nature at high altitudes (70 and 80 km).

Conclusions

In this study, two sub-tropical stations situated at Mt. Abu in the NH and Reunion Island in the SH are selected to distinguish the characteristics of middle atmosphere in different hemispheres. For this purpose, the SABER onboard TIMED satellite observations are used for thermal structure in stratosphere-mesosphere system during January 2002 to September 2015. Temporal variation of SABER observations in this period shows that stratopause temperature is warmer in Mt. Abu as compared to Reunion Island. Moreover, mean climatological behaviour of upper mesosphere shows that NH has higher temperature as compared to SH. Monthly climatology based on 14 years SABER observations shows that stratopause is warmer in NH, and cooler in upper mesosphere, as compared to SH. In general, stratosphere shows higher temperature over Mt. Abu compared to Reunion Island. The stratosphere sudden warming which are triggered from the NH Polar Regions and influence different latitudes depending upon their strength may be one of the reason for this anomalous behaviour. Monthly temperature anomaly shows that both hemisphere show identical temperature structure in stratosphere, with small positive anomaly at lower mesosphere in SH as compared to NH for the months of December to February. Mean temperature values during winter and summer season show that NH is warmer and cooler in summer and winter season respectively as compared to SH in stratosphere. Upper mesosphere shows a reverse nature. Three atmospheric models are also used to compare and contrast the vertical thermal structure for different months over both regions. Temperature simulated from these models matches well with SABER observations for lower stratosphere. Significant discrepancies are also seen in the mesospheric heights and differences are more pronounced over Reunion Island in the SH.

In addition to this, seasonal oscillations are studied using LSP and Wavelet transform techniques for stratospheremesosphere system which will help to distinguish the wave dynamics or wave features over these regions in the NH and SH. Results show that AO are stronger over Reunion Island at 20 km. The AO are found at both selected locations for all altitudes except 50 and 60 km. The QBO are dominant at 20-40 km over Mt. Abu region, and are exist at 40 km in SH. Overall, results suggest that AO are dominated in NH and SH as compared to SAO and QBO. The strength of AO is nearly similar in stratosphere, but strength of AO are more in NH as compared to SH in upper mesosphere. However, SAO are stronger in SH as compared to NH. Wavelet analyses found a very interesting results for AO at 80 km, which shows that AO are significant over Mt. Abu region, but are not dominated throughout the study period over the SH. However, SAO are more stronger in southern region.

Fig. 1 Fig. 3

 13 Fig. 1 Geographic locations of the study regions

Fig. 5

 5 Fig. 5 Thermal structure of Mt. Abu and Reunion Island in winter and summer seasons. Differences between Mt. Abu and Reunion Island temperature are also shown in right panels

Fig. 7

 7 Fig. 7 Same as Fig. 6 but for Reunion Island

Fig. 8

 8 Fig. 8 Lomb Scargle periodogram for SABER retrieved temperature at every 10 km. The black dotted line shows significant changes at 95 % confidence interval

Fig. 10

 10 Fig. 10 Same as Fig. 9 but for Reunion Island

Table 1

 1 SABER observed and atmospheric models simulated temperature over Mt. Abu

	Altitude (km) SABER (K) MSIS (K) NRLMSIS (K) CIRA (K) SABER-MSISE (K) SABER-NRLMSIS (K) SABER-CIRA (K)
	20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0	205.2 229.1 250.8 259.9 238.6 211.3 193.7 192.3	209.7 229.5 256.8 264.2 239.8 215.0 198.6 178.7	209.7 229.5 256.9 263.8 241.8 220.5 197.7 173.8	210.2 229.5 256.0 266.0 241.3 216.8 205.5 188.2	-4.5 -0.3 -6.0 -4.4 -1.2 -3.7 -4.9 13.6	-4.5 -0.3 -6.2 -3.9 -3.2 -9.2 -3.9 18.5	-5.0 -0.4 -5.2 -6.1 -2.7 -5.5 -11.7 4.1
	100.0	183.7	173.5	179.8	186.9	10.3	3.9	-3.1
	Table 2 SABER observed and atmospheric models simulated temperature over Reunion Island
	Altitude (km) SABER (K) MSIS (K) NRLMSIS (K) CIRA (K) SABER-MSISE (K) SABER-NRLMSIS (K) SABER-CIRA (K)
	20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0	205.0 229.2 250.8 260.7 239.8 210.2 195.1 191.4	208.7 229.7 257.1 264.9 240.6 214.6 197.1 178.3	208.7 229.7 257.2 264.6 242.2 219.0 197.4 174.4	209.1 229.7 256.7 267.0 242.4 214.9 204.3 189.1	-3.7 -0.5 -6.3 -4.2 -0.8 -4.4 -2.0 13.2	-3.7 -0.5 -6.4 -3.9 -2.4 -8.8 -2.2 17.0	-4.1 -0.5 -5.9 -6.3 -2.6 -4.7 -9.2 2.3
	100.0	182.7	176.1	176.9	187.0	6.6	5.7	-4.3

Acknowledgments Authors are thankful to SABER onboard TIMED, CIRA-86, MSISE-90 and NRLMSIS-00 team members to provide valuable temperature data. This work is supported by Dept. of Space, Govt. of India.