André Vítor

Santos Simõis

Lucas Kaique

Martins Roncaselli

Vinícius Jessé

Rodrigues De Oliveira

Maria Eduarda Rocha

Santos Medina

Hasina H Ramanitra

M Stephen

Deuber L Silva Agostini

Roger C Hiorns

Clarissa De

Almeida Olivati

Rodrigues De

Polyfullerene Thin Films Applied as NH 3 Sensors

Keywords: Fullerene, NH 3, Gas sensors

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1.

Introduction

The ease and low cost of processing has made polymeric materials abundantly present in our daily lives. In the electrical industry, polymeric materials were initially used in order to replace paper-based insulators. Their acceptance was rapid because they are light, cheap and highly insulating materials 1 .

The discovery of buckminsterfullerene (C 60 ) in 1985, more commonly known as fullerene, occurred accidentally in an attempt to simulate the conditions of nucleation of carbon atoms in cold red giant N-type stars 2 . Because of its structural stability, C 60 is considered to be the third carbon allotrope, after diamond and graphite 3 .

One of the problems associated with fullerene is its very strong tendency to aggregate in an uncontrolled manner. This means that its sensor, electronic and medical properties can be hard to exploit. A route around this problem can be provided by incorporating fullerene into a polymeric structure. This means that its aggregation becomes more controlled, its solubility in common solvents is improved, and as a material it becomes more malleable and easier to handle. One of the simplest ways of incorporating fullerene into a polymer is to use it like a monomer and make it part of the polymer 's back-bone. This can be done via several routes, notably by copolymerization with comonomers via radical 4 or cycloaddition 5 routes. These procedures can result in the production of high yields of oligo(fullerene)s and poly(fullerene)s that contain high proportions of fullerene (typically > 60% by weight).

Gas sensors are mostly composed by p-type materials, such as P3HT, however, n-type materials have been relatively ignored and are not so well understood 6 . Fullerenes are of particular interest due to their structural stability and their exceptional affinity for electrons 7,8 .

The study of the interaction between gas molecules and semiconductors is one of the most studied areas and has attracted the attention of many researchers. Monitoring gases play a high importance role, especially toxic gases of our daily life. Gas sensors are also commonly used as safety devices, especially in industrial areas where toxic chemical compounds might leak. One example is NH 3 which is an important part of the atmosphere and plays the role of neutralizing acid gases and maintaining the ecological balance 9 . However, NH 3 is also a toxic gas when at higher concentrations, especially in industrial processes and biological metabolisms 10 .

In this work, different poly(fullerenes) and oligo (fullerenes) were studied as thin films, made using the drop-casting deposition technique. This method was chosen as it is simple, fast and of low-cost 11 . It consists of simply dripping a solution of the material under study on to the substrate surface, and then evaporation the solvent. The film remains on the substrate due to Van der Waals force 12,13 . The drop casting technique, however, does not allow control over the formation of the film, causing the formation of uncontrolled agglomerates, resulting in films with a heterogeneous surface. Lack of control over film formation also limits control over film thickness. Some factors such as deposited volume and solution concentration can help regulate the final film thickness, but with low accuracy 11,14 .

Methodology

Materials

In this work, three n-type materials were studied, being, phenyl-C 61 -butyric acid methyl ester (PCBM), oligo {(phenyl-C 61 -butyric acid methyl ester)-alt-[1,4-bis(bromomethyl)-2,5bis(octyloxy)benzene]} (OPCBMMB) and poly {[bispyrrolidino(phenyl-C 61 -butyric acid methyl ester)]-alt-[2,5-bis(octyloxy)benzene]} (PPCBMB). Figure 1 To make the thin films by drop casting deposition, solutions were made from these materials, using chloroform as the solvent at a concentration of 1.0 mg mL -1 .

Drop casting films

Thin films based on the Drop casting technique were prepared using the polymers under study with a concentration of 0.2 mg mL -1 , having chloroform as solvent. Using an electronic pipette, the solution of each material already homogenized was spread over the solid substrate on a table surface, and after 40 minutes of evaporation of the volatile solvent in a controlled environment of 22°C, a layer of the polymer could be obtained in the form of film. The solvent evaporation process is an essential part of the deposition technique, where we have the option to choose between letting the solvent evaporate naturally or use thermal processes to accelerate the process. However, by disturbing the system, we create changes in the final morphology of the film, as stated by Kumar et al. 14 . In this work, we opted to let the solvent evaporate naturally. Figure 2 shows a representation of this process.

DC electrical characterization

The measurements of electrical characterizations in direct current (DC) and current by time (I vs t) were performed using a Keithley 238 voltage source.

After deposition of the films on Interdigitated gold electrodes (IDE-Au) that has the characteristics of: N = 50 digits with dimensions of 110 mm high (h), 8 mm long (L) and 100 μm wide (w), as shown in Figure 3.

The films were subjected to current by voltage measurements (I vs V), varying the applied potential from -15 to 15 Volts, before and after measurements with NH 3 vapor, to verify the electrical conductivity of the material and the electrochemical effect of the interaction. The measurements of current over time with the same samples are the stage in which the films were exposed to the steam flow, which in contact with the films generated an electrical response, characterizing them as a gas sensor.

Gas sensors

The sensory measurements were made through the study of current versus time curves (I vs t). In the controlled gas measurement system, an inert nitrogen (N 2 ) flow is initially applied to stabilize the sample and create a baseline.

After this initial stabilization, the NH 3 flux is released by drag with the same N 2 , now the ammonia flux acting on the sensor substrate, an interaction that generates a change in the electrical current registered in the device. This process between the flow of NH 3 vapor and pure N 2 (inert gas) is alternated so that a visible electrical interaction is observed. The process is alternated in cycles of 10 minutes at a constant flow of 60N.L/h. Figure 4 represents a schematic of the system used for the sensory measurements.

Results and Discussions

Sensor responses

The experiment was carried out in cycles of 10 minutes, alternating between the baseline of N 2 and NH 3 . During the entire process, the samples were subjected to a constant The graphs reveals that the three materials responded to the presence of NH 3 , showing a large increase in the measured current. The highest response is observed on PCBM, which had a peak of 57 μ A, followed by OPCBMMB, with a peak of 0.37 μ A. PPCBMB showed the lowest peak, 78 nA.

As these materials have different size (PCBM, oligo-fullerene and poly-fullerene), it is possible to observe variations in the behavior of the structures of each material, and after being exposed to ammonia vapor, the PCBM films showed a fast response in contact with the analyte, decreasing electrical stimulation over time.

As for oligo and poly fullerenes, we observed a different response from PCBM. OPCBMMB showed an increase in electrical stimulus at each cycle, while PPCBMB showed an apparently continuous response without major changes during exposure cycles. This performance in gas response for the three materials can be explained by the fact that they PCBM in common in their structure and that they are n-type materials, which reacts favorably with NH 3 , increasing the semiconductor nature of the material, revealing current peaks when exposed to gas 19,20 , we verified in the ammonia responses that the oligo and poly fullerenes have a more consistent response to each cycle, keeping the current at close values, according to Table 1. Interestingly, although the response is less marked in the first cycles, OPCBMMB showed a more consistent response at 60 and 80 minutes (3 rd and 4 th cycles).

Another important factor is the technique used in this work. The Drop casting technique has a low level of control in the materials organization, which results in a film with low homogeneity, as stated in previous works from Kanoun 11 ,Pimentel 12 and Castro 21 . This can create void on the structure, capable of absorbing NH 3 molecules by weak chemical interaction 20 . The materials then capture the gas charges, which then participate in the conduction process 22,23 . Observing the graphs, it is also possible to verify that, not only the materials responded to the NH 3 , but also this response is reproducible. The materials showed the ability to repeat the gas response, indicating that sensor devices made from them can be reused 24,25 .

DC characterization

Figure 6 shows I vs V characteristic curves. The samples were subjected to a voltage ranging from -15 V to 15 V, with a 0.5 V step. The process was done before and after exposure to NH 3 , in order to observe possible changes in the electrical behavior of the materials, and to check if the process is reversible.

The linear curves of current (I) vs. voltage (V), represented in Figure 6 show that the materials presented an ohmic behavior, due to contact with the Au/film/Au configuration 26 , especially in the lower voltage regions. Ohmic contacts, such as those observed, have the characteristic of not influencing the density of carriers in the studied material volume when an electrical voltage is applied 26,27 . This feature allows us to obtain information about these materials, such as electrical conductivity 28 . This observed behavior can be described by the linear equation (y = ax b)+, with b = 0 . From Ohm's Law, with Equations 1 and 2, it is possible to determine the resistance (R) and the electrical conductivity (σ ), since the conductivity is the inverse of resistivity ( ρ ), ( σ =1/ ρ ).

V = R.i (1) R = K.ρ (2)
In Equation 2, K is the cell constant, defined by the geometry of the used IDE. In this work, the IDE used has a cell constant with a value of 5.1 m -1 , as defined in previous works 18 , following the model of Olthuis et al. [START_REF] Olthuis | Planar interdigitated electrolyteconductivity sensors on an insulating substrate covered with Ta2O5[END_REF] . Table 2 disposes the conductivity values to the materials before and after exposure to NH 3 .

Observing the graphs, it is possible to state that the three materials kept their ohmic behavior. However, although OPCBMMB and PPCBMB maintained similar conductivity values, PCBM showed a considerable reduction to its value. This might be due to the tendency of fullerenes to react permanently with amines, which would explain the behavior of PCBM 30 . The oligomer and polymer probably react less due to the steric blocking of permanent reactions by the comonomers in the polymer structures, thereby maintaining their structures and sensitivity to ammonia over time. These reactions may change with light and this will be part of a future study.

When analyzing the data in Table 2, we verified that before exposure to the gas, all materials have low conductivity, with PCBM presenting the highest conductivity value among them 31,32 , about 1 order of magnitude higher. When exposed to gas, we found that the structure of pure PCBM, when in contact with NH 3 vapor, decreased its electrical conductivity by an order of magnitude, due to a possible nucleophilic reaction of NH 3 with PCBM, reducing its ability to exchange electrical charges, which can be seen in Figure 5, observing the drop in current peak over time. The other materials did not undergo major changes, maintaining their similar conductivities before and after exposure to the gas. Both OPCBMMB and PPCBMB have ramifications that can block the permanent chemistry with the gas, which reduces interactions with ammonia, but allows a response reproducibility over time, a fact that can be compared to PCBM in Figure 5, where these materials showed consistent results throughout each cycle 33,34 .

Conclusions

Analyzing the results obtained from this work, we observe the behavior of PCBM, OPCBMMB and PPCBMB when immersed in an ammonia atmosphere, verifying the electrical response generated by the gas stimulus. It is noted that PCBM, despite having the highest peak among the materials studied, shows a drop in its electrical response at each cycle, due to a possible nucleophilic reaction between the gas and the material, which reduces its ability to exchange electrical charges. This phenomenon is not observed in OPCBMMB and PPCBMB, as they present ramification capable of blocking these permanent reactions with the gas, which reduces their interaction capacity, resulting in smaller, but more consistent response peaks.

By analyzing the conductivity curves (I vs V) before and after the interaction with ammonia, it can be observed that the three materials studied preserved their ohmic behavior after ammonia stimuli, with OPCBMMB and PPCBMB maintaining very close conductivity values, while PCBM showed a significant reduction in its electrical conductivity, about an order of magnitude, but also maintaining its ohmic behavior

The results obtained shows that these materials present favorable conditions for the production of NH 3 sensors, as they all react to the gas stimulus, and are capable of reproducing the response, without losing their electrical properties. In this context, we highlight OPCBMMB and PPCBMB, as they are able to reproduce the results with better consistency.

  represents the chemical structure of the studied materials. PCBM was obtained from Sigma Aldrich, OPCBMMB was prepared as detailed by Ramanitra et al. 15 , and PPCBMB was prepared as indicated by Stephen et al. 16 .

Figure 1 .

 1 Figure 1. Chemical structure of a) PCBM; b) OPCBMMB; c) PPCBMB.

Figure 2 .

 2 Figure 2. Scheme showing the Drop casting deposition process 11 .

Figure 3 .

 3 Figure 3. Representation of a golden interdigitate electrode with N = 10 17 .

Figure 4 .

 4 Figure 4. Representation of the sensory measurements system 18 .

Figure 5 .

 5 Figure 5. I vs t curves for the sensor devices with Drop casting films of: a) PCBM; b) OPCBMMB; c) PPCBMB.

- 10 Figure 6 .

 106 Figure 6. I vs V curves before and after exposure to NH 3 for: a) PCBM; b) OPCBMMB; c) PPCBMB.

Table 1 .

 1 Current peaks observed at each cycle for the studied materials

Table 2 .

 2 Material conductivities before and after exposure to NH3.

Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçoamentode Pessoal de Nivel Superior -Brasil (CAPES) -Finance Code 001.


Penny Haxell
email: pehaxell@uwaterloo.ca

Reza Naserasr

A note on ∆-critical graphs

A k-critical graph is a k-chromatic graph whose proper subgraphs are all (k -1)colourable. An old open problem due to Borodin and Kostochka asserts that for k ≥ 9, no k-critical graph G with k = ∆(G) exists, where ∆(G) denotes the maximum degree of G. We show that if a certain special list-colouring property holds for every 8-critical graph with ∆ = 8 (which is true for the apparently only known example) then the Borodin-Kostochka Conjecture holds. We also briefly survey constructions of ∆-critical graphs with ∆ ≤ 8, highlighting the apparent scarcity of such graphs once ∆ exceeds 6.

Introduction

A famous open problem about extending Brooks' Theorem [START_REF] Brooks | On colouring the nodes of a network[END_REF] was posed by Borodin and Kostochka [START_REF] Borodin | On an upper bound of a graph's chromatic number depending on the graph's degree and density[END_REF] in 1977. It can be conveniently formulated as follows. Here ∆ denotes the maximum degree of a graph, and by a k-critical graph we mean a graph G with the property that the chromatic number χ(G) = k but χ(G ) ≤ k -1 for every proper subgraph G of G. Conjecture 1. For each k ≥ 9, the only k-critical graph with ∆ ≤ k is K k .

Since ∆(K k ) = k -1, this is equivalent to asserting the non-existence of ∆-critical graphs once ∆ ≥ 9. Phrased in similar language, Brooks' Theorem implies that for each k ≥ 4, the only k-critical graph with ∆ + 1 ≤ k is K k . The condition k ≥ 9 is necessary, due in particular to the graph C 5 [K 3 ] shown in Figure 1.

Much progress has been made towards a solution to the Borodin-Kostochka Conjecture, for example [START_REF] Borodin | On an upper bound of a graph's chromatic number depending on the graph's degree and density[END_REF][START_REF] Cranston | Coloring claw-free graphs with ∆ -1 colors[END_REF][START_REF] Cranston | Graphs with χ = ∆ have big cliques[END_REF][START_REF] Cranston | Coloring a graph with ∆ -1 colors: Conjectures equivalent to the Borodin-Kostochka Conjecture that appear weaker[END_REF][START_REF] Gupta | Borodin-Kostochka's Conjecture on (P 5 , C 4 )-free graphs[END_REF][START_REF] Kostochka | Degree, density and chromatic number of graphs[END_REF][START_REF] Mozhan | Chromatic number of graphs with a density that does not exceed twothirds of the maximal degree, Metody Diskretn[END_REF][START_REF] Reed | A Strengthening of Brooks' Theorem[END_REF] (see e.g. [START_REF] Cranston | Graphs with χ = ∆ have big cliques[END_REF] for a detailed description of previous work). Despite these developments, the conjecture remains open in general. Here we mention just two of the strongest results to date, namely, that the conjecture holds provided ∆ is a sufficiently large (unspecified) constant [START_REF] Reed | A Strengthening of Brooks' Theorem[END_REF], and that any ∆-chromatic graph with ∆ ≥ 13 must contain K ∆-3 [START_REF] Cranston | Graphs with χ = ∆ have big cliques[END_REF]. In [START_REF] Cranston | Coloring a graph with ∆ -1 colors: Conjectures equivalent to the Borodin-Kostochka Conjecture that appear weaker[END_REF] it was shown that the conjecture is equivalent to the seemingly much weaker assertion that every ∆-chromatic graph with ∆ ≥ 9 contains the graph obtained from K ∆ by removing the edges of a K ∆-3 .

One simple but important step in addressing Conjecture 1, shown in [START_REF] Kostochka | Degree, density and chromatic number of graphs[END_REF][START_REF] Catlin | Embedding subgraphs and coloring graphs under extremal degree conditions[END_REF], has been to reduce the general problem to graphs with ∆ = 9. (Similar arguments were used in e.g. [START_REF] Cranston | Graphs with χ = ∆ have big cliques[END_REF][START_REF] Cranston | Coloring a graph with ∆ -1 colors: Conjectures equivalent to the Borodin-Kostochka Conjecture that appear weaker[END_REF][START_REF] Mozhan | Chromatic number of graphs with a density that does not exceed twothirds of the maximal degree, Metody Diskretn[END_REF] as well.) In this note we continue the same theme, linking the general conjecture to the behaviour of graphs with ∆ ≤ 8.

We will refer to a particular restricted notion of list colouring, called non-identical list colouring (abbreviated NIL), defined as follows. For a set S, an integer k, and a graph G, an (S, k)-NIL assignment of G is an assignment

L of lists L(v) ⊆ S to each vertex v ∈ V (G) such that each |L(v)| = k, and L(v) = L(w) for some pair v, w ∈ V (G). We say that G is (S, k)-NIL colourable if every (S, k)-NIL assignment L of G admits an L-list colouring of G.
For example, it is easy to verify (see Section 2) that the complete graph K n is (S, n -1)-NIL colourable for every S.

The main result of this note is the following, where [n] denotes {1, . . . , n}. The proof appears in Section 3.

Theorem 2. Suppose that every 8-critical graph with ∆ ≤ 8 is regular and ([8], 7)-NIL colourable. Then the only 9-critical graph with ∆ ≤ 9 is K 9 (i.e. the Borodin-Kostochka Conjecture holds).

Recall that the lexicographic product of graphs G and H is the graph G[H] with vertex set V (G) × V (H), in which (x, u) is adjacent to (y, v) whenever xy ∈ E(G), or x = y and uv ∈ E(H). The (regular) 8-critical graph C 5 [K 3 ] (see Figure 1) is shown in Section 2 to be ([8], 7)-NIL colourable. Since we are unaware of any 8-critical graphs with ∆ ≤ 8 besides C 5 [K 3 ] and K 8 , the following corollary tempts us to conjecture that there are no others.

Corollary 3. Suppose that the only 8-critical graphs with ∆ ≤ 8 are C 5 [K 3 ] and K 8 . Then the Borodin-Kostochka Conjecture holds.

We end this brief note with a discussion on constructing ∆-critical graphs for small ∆ in Section 4. In particular, we point out there that while infinite families of such graphs exist when ∆ ≤ 6, it seems more difficult to construct examples for ∆ = 7, and (as mentioned) even more so for ∆ = 8.

Non-identical list colouring

In this section we verify that the graph

C 5 [K 3 ] shown in Figure 1 is ([8], 7)-NIL colourable. Observe that C 5 [K 3 ]
has 15 vertices and independence number 2, so it is not 7-colourable. An 8-colouring can be easily found (χ(C 5 [K 3 ]) ≤ 8 also follows from Brooks' theorem). To see that it is critical, observe that the graph is vertex transitive and there are only two types of edges (those in triangles x i y i z i and those connecting them). Given an [START_REF] Gupta | Borodin-Kostochka's Conjecture on (P 5 , C 4 )-free graphs[END_REF]colouring φ, let us assume that 8 is the unique colour that appears exactly once, and that φ(x 1 ) = 8. By symmetry, we may assume that φ(z 2 ) = φ(z 5 ) = 7, and that colors 1, 2, . . . , 6 each appear on neighbours of x 1 exactly once. Then in a subgraph missing x 1 y 1 we may recolour x 1 to φ(y 1 ), and in a subgraph missing x 1 x 2 we may recolour x 1 to φ(x 2 ). This shows that C 5 [K 3 ] is also 8-critical.

x 1 x 2 x 3 x 4 x 5 y 1 y 2 y 3 y 4 y 5 z 1 z 2 z 3 z 4 z 5 Figure 1: C 5 [K 3 ]
First we note the following simple consequence of the well-known fact (equivalent to Hall's Theorem) that a collection of sets

{L(v) : v ∈ V } has a system of distinct representatives {s v ∈ L(v) : v ∈ V } if and only if | v∈T L(v)| ≥ |T | for each T ⊆ V .
Lemma 4. For every n ≥ 2 and every set S, the complete graph

K n is (S, n -1)-NIL colourable. Lemma 5. The graph C 5 [K 3 ] is ([8], 7)-NIL colourable. Proof. Fix an ([8], 7)-NIL assignment L of G = C 5 [K 3 ]
. By symmetry, we may assume that two (adjacent) vertices with different lists are among x 1 , x 2 , y 1 , y 2 . Let K denote the K 4 induced by these four vertices. Since every pair of vertices of G have at least 6 common colours in their lists, by renaming colours if necessary we may choose the following partial colouring ϕ:

• ϕ(z 1 ) = ϕ(z 3 ) = 7. • ϕ(z 2 ) = ϕ(z 5 ) = 8. • ϕ(x 3 ) = ϕ(x 5 ) = 6 and ϕ(y 3 ) = ϕ(y 5 ) = 5. • For each v ∈ K ∪ {x 4 , y 4 , z 4 } set L (v) = L(v) \ {5, 6, 7, 8}. Then we may easily colour each vertex v ∈ {x 4 , y 4 , z 4 } from L (v) since |L (v)| ≥ 3 for each.
It remains to colour the vertices v of K from their current lists L (v). If |L (v)| = 3 for each v ∈ K, then each L(v) contains all of {5, 6, 7, 8} and hence, by our choice of K, we find that

{L (v) : v ∈ K} is a ([4], 3)-NIL assignment of K 4 . If |L (x)| ≥ 4 for some x ∈ K,
then we may assign to each v in K a 3-subset of L (v) such that not all are identical. Hence, in either case, by Lemma 4, we may complete the colouring of K and hence of G.

Proof of Theorem 2

As mentioned in the Introduction, and shown in [START_REF] Kostochka | Degree, density and chromatic number of graphs[END_REF][START_REF] Catlin | Embedding subgraphs and coloring graphs under extremal degree conditions[END_REF] (see also [START_REF] Cranston | Graphs with χ = ∆ have big cliques[END_REF][START_REF] Cranston | Coloring a graph with ∆ -1 colors: Conjectures equivalent to the Borodin-Kostochka Conjecture that appear weaker[END_REF][START_REF] Mozhan | Chromatic number of graphs with a density that does not exceed twothirds of the maximal degree, Metody Diskretn[END_REF]), the Borodin-Kostochka Conjecture can be reduced to the specific case of k = 9. A useful tool for such purposes is the following result of King [START_REF] King | Hitting all maximum cliques with a stable set using lopsided independent transversals[END_REF] (which is based on [START_REF] Haxell | A note on vertex list-colouring[END_REF]). Here ω(G) denotes the maximum size of a clique in G.

Theorem 6. If a graph G satisfies ω(G) > 2 3 (∆(G) + 1), then G contains an independent set I such that ω(G -I) = ω(G) -1.
Our proof of Theorem 2 begins with an application of Theorem 6, which in this context is quite standard. For example it essentially repeats the proof of Lemma 1.8 in [START_REF] Cranston | Coloring a graph with ∆ -1 colors: Conjectures equivalent to the Borodin-Kostochka Conjecture that appear weaker[END_REF], but we include this argument in order to establish how the 8-critical graph H sits in G. This is needed for the rest of the proof.

We may now begin the proof of Theorem 2. Suppose that there exist 9-critical graphs G with ∆(G) ≤ 9 that are distinct from K 9 , and choose one such G with the smallest number of vertices. Then by Brooks' Theorem we know ∆(G) = 9, since otherwise G would be 8-colourable. By Theorem 6 we may choose a maximal independent set I in G that intersects every 8-clique. (Note that if ω(G) ≤ 7 then any maximal independent set will do.)

By maximality of I we know ∆(G -I) ≤ 8, by 9-criticality of G we know χ(G -I) ≤ 8 (and hence χ(G-I) = 8), and by choice of I we have ω(G-I) ≤ 7. Let H be an 8-critical subgraph of G -I. Then clearly we have ∆(H) ≤ 8 and ω(H) ≤ 7 as well. Hence in particular H is not K 8 , so again by Brooks' Theorem ∆(H) = 8. Thus, by the assumption of the theorem, H is 8-regular and ([8], 7)-NIL colourable.

We denote by I H the subset I ∩ N G (H). Since ∆(G) = 9 and H is 8-regular, by maximality of I we know d I H (v) = 1 for each vertex v of H, and I H is a vertex cut in G separating H from G -I -H. Furthermore since H is 8-regular and not K 9 it has more than 9 vertices, so |I H | ≥ 2.

By 9-criticality of G we know that G-H is 8-colourable. First suppose that G-H has an 8-colouring φ in which two vertices of I H receive different colours. Since each vertex v of H has exactly one neighbour v I in I H , the list assignment L given by L(v) = [START_REF] Gupta | Borodin-Kostochka's Conjecture on (P 5 , C 4 )-free graphs[END_REF]\{φ(v I )} is an ([8], 7)-NIL assignment for H. Hence, by the assumption, H has an L-colouring, which together with φ shows that G is 8-colourable, giving a contradiction. Thus we may assume that every 8-colouring of G -H must give all vertices of I H the same colour. In other words, adding any edge xy to G -H where x, y ∈ I H results in a 9-chromatic graph G -H + xy. For four vertices u i ∈ I H , 1 ≤ i ≤ 4, u i = v, by Claim 1, we have copies K vu i of K - 9 in G, each of which contains v. Since d(v) ≤ ∆(G) = 9, these four subgraphs have another common vertex in G -I -H, say w. Then w has eight neighbours in K vu 1 together with u 2 , u 3 and u 4 , thus there are a total of at least 11 neighbours of w in G, contradicting the fact that ∆(G) = 9. This completes the proof of Theorem 2.

∆-critical graphs for small ∆

The graph C 5 [K 3 ] is one example of a critical graph obtained by blowing up each vertex of an odd cycle into a clique. One can easily construct infinite families of k-critical graphs with ∆ = k for k = 4, 5 by using such blow-ups using clique sizes 1 and 2.

Another simple construction of critical graphs from [START_REF] Catlin | Embedding subgraphs and coloring graphs under extremal degree conditions[END_REF] (generalizing examples from [START_REF] Benedict | On graphs having prescribed clique number, chromatic number, and maximum degree[END_REF]) is as follows. Fix k ≥ 4, let G be a graph with ∆(G) ≤ k, and suppose G has a vertex x of degree k -1. Form a new graph G k by "evenly splitting" x into 3 vertices (i.e. remove x and add an independent set I of three vertices whose degrees differ by at most one and whose neighbourhoods partition N (x)), and joining each vertex of

I to a new clique K of size k -2. It is straightforward to verify that G k is k-critical if and only if G is k-critical. Since each v ∈ V (G) ∩ V (G k ) has degree d G (v) in G k , each v ∈ K has degree (k -3) + 3 = k, and each v ∈ I has degree k -2 + k-1 3 or k -2 + k-1 3 , we see that if k -2 + k-1 3 ≤ k then ∆(G k )) = k. This holds for k = 4, 5, 6, 7. Moreover if k -2 + k-1 3 ≤ k -1, then G k
has a vertex of degree less than k, which is true for k = 4, 5, 6. Hence for these values, this construction may be repeated an arbitrary number of times, starting e.g. with K k , giving an infinite family of k-critical graphs with ∆ = k. For k = 7, however, since no new vertex of degree k -1 is introduced by the construction, if we start with K 7 the process will produce 7-critical graphs with ∆ = 7 on 7i vertices for i = 1, 2, . . . 8, thus terminating with graphs on 56 vertices. This operation produces various nonisomorphic 7-critical graphs on 7i vertices for i = 3, . . . , 8. For example, for such graphs on 56 vertices, we note that the subgraph J induced on the set of 21 split vertices is a 2-regular graph where there is (exactly) one edge connecting each pair of triples representing two vertices of K 7 . In particular, for any partition F 1 , . . . , F s of E(K 7 ) into connected even-degree subgraphs of K 7 , we may construct J whose components are cycles of lengths |E(F 1 )|, . . . , |E(F s )|, by following an Euler tour in each of the F i . There are many such partitions, for example a partition of E(K 7 ) into three 7-cycles, or a partition into seven 3-cycles (the lines of Fano plane), or the whole set E(K 7 ) as a single partition class (for which J is a 21-cycle).

Since the graph obtained from C 5 [K 3 ] by removing two nonadjacent vertices is also 7-critical, another family of 7-critical graphs with ∆ = 7 can be built from it using the above operation. This gives further examples, with vertex sets of sizes 20, 27, and 34. To the best of our knowledge this desribes all the known 7-critical graphs satisfying ∆ = 7. Thus we pose the following natural questions. Since G k is k-critical if and only if G is k-critical, one potential way of showing that the above list is complete for ∆ = 7 might be to show that every 7-critical graph with ∆ = 7, aside from C 5 [K 3 ]-I, contains the graph obtained from K 8 by removing the edges of a triangle.

Finally we remark that further examples of ∆-critical graphs for ∆ = 4, 6 that are not of either of the types described above are given by the square C 2 8 of the 8-cycle and the cube C 3 11 of the 11-cycle respectively (the latter is shown in Figure 2). More generally, the rth power C r 3r+2 of the (3r + 2)-cycle forms the basis for a family of tight examples for a conjecture of Reed [14], that is closely related to Conjecture 1. This conjecture proposes that for every graph G one has χ(G) ≤ ∆(G)+ω(G)+1

2

. The graph C r 3r+2 [K t ] has (3r + 2)t vertices, it is ∆ = ((2r + 1)t -1)-regular, it has clique number ω = (r + 1)t, and independence number 2. This implies that its chromatic number is at least (3r + 1)t 2 = (2r + 1)t -1) + (r + 1)t + 1 2 ∆(G) + ω(G) + 1 2 .

It is not difficult to give a colouring using only this many colours. The well-known example C 5 [K t ] is the special case r = 1.

Claim 1 .

 1 Every pair x, y ∈ I H of distinct vertices lies in a K - 9 -subgraph K xy of G -H with V (K xy ) ∩ I H = {x, y}. (Here K - 9 denotes the graph obtained by removing one edge from K 9 .) To verify the Claim, observe that since G -H + xy is 9-chromatic, it contains a 9critical subgraph J. Since |V (J)| ≤ |V (G -H)| < |V (G)| and ∆(J) ≤ ∆(G) = 9, by minimality of our counterexample G, we conclude that J = K 9 . Clearly J must contain the edge xy, so J -xy is the claimed K - 9 . Recalling that |I H | ≥ 2 and ∆(G) = 9, Claim 1 immediately implies a contradiction if any v ∈ I H satisfies d H (v) ≥ 3. Hence we may assume d H (v) ≤ 2 for each v ∈ I H . We know H has more than 9 vertices, so since d I H (u) = 1 for each vertex u of H, we find that |I H | ≥ 5. Fix v ∈ I H .

Problem 7 .

 7 Does there exist an 8-critical graph with ∆ = 8 different from C 5 [K 3 ]? Problem 8. Does there exist an infinite family of 7-critical graphs with ∆ = 7?

Figure 2 : C 3 11

 211 Figure 2: C 3 11

* Partially supported by NSERC † This work is supported by the ANR project HOSIGRA (ANR-17-CE40-0022).

Statements & Declarations

The work is supported by NSERC (Canada) and by the ANR (France) project HOSIGRA (ANR-17-CE40-0022).

The authors have no relevant financial or non-financial interests to disclose.

Both authors contributed to the study conception and design and the writing. Both authors read and approved the final manuscript.


Min Feng
email: min.feng@univ-lyon3.fr

Bourazzouq
email: driss.bourazzouq@uvsq.fr

Driss

Solutions for Coping with Technological Advancement: A Case Study. ST-AIMS 13 : Pratiques d'anticipation et environnements extrêmes

Keywords: technostress, coping strategy, interaction, team manager, interdependence, leader member exchange (LMX)

Given the information overload and technostress that affects team managers within the information and communication technology management framework, this study examines coping strategies and behaviors adopted by team managers to address technostress. Moreover, it evaluates the choice of coping strategies to improve performance. Using the concepts of interdependence and leader member exchange (which refers to the identification of coping strategies), a study involving three companies was conducted to identify coping strategies. Overall, four interactional coping strategies were identified: non-procedural strategy, professional support strategy, achievement mechanism strategy, and mutual trust strategy. Four new coping theories were created based on these strategies to address technostress from an international perspective, thus enriching the literature on coping strategies and technostress. Empirically, the results explain a wide range of team managers' behaviors. These theories and strategies can be adapted as a solution for coping with technostress.

Solutions for Coping and Responding to Technologies

Enhancement: A Study of Three Companies

INTRODUCTION

Previous studies (Feng, 2020) suggest that the ubiquitous use of Information and Communication Technologies (ICTs) may create technostress that diminishes the performance of the Proximity Manager (PM) when stress levels increase with the role. Companies are implementing coping strategies to address technostress. Initially, these coping strategies were a result of management considering the opinions of employees and experts. They were then shared with team managers (TMs) with the aim of determining adequate and operational adaptive strategies. Finally, employees in each department applied these strategies. In this sense, it is simply interactions between organizations and employees; that is, exchanges, coordination, cooperation, and communication between employees and TM.

We sought to study the adaptive behaviors of TMs in the fight against technostress and its increase with ICT. Theoretically, our research was conducted in ICT management to analyze how information overload and technostress affected local managers (PMs) by combining the concepts of interdependence and LMX (exchanges between manager and employee). The aim was to identify coping strategies and to understand the behaviors adopted by PMs in the face of the technostress phenomenon. Empirically, our study was conducted to identify different types of coping (adaptive) strategies. We adopted a policy of interactional analysis involving the managers and their subordinates. The research questions were as follows:

• RQ 1: What adaptive and coping strategies and solutions are taken by TMs to manage technostress in an interactional environment?

Our study is fundamental in this digital era where ICT is integrated into almost all fields and sectors. [START_REF] Bag | Big data analytics as an operational excellence approach to enhance sustainable supply chain performance[END_REF] studied how digitalization has enhanced performance in procurement, remanufacturing operations, and supply chains. Our research was conducted through observation and interviews. We conducted a literature review prior to and along with the empirical research, followed by a comparison of the theory and empirical results. This research brings novelty in dealing with ICT-technostress by providing adequate and adaptable coping solutions for all types of situations. We have a micro foundation vision (Barro, 1993) in our Annecy, 31 mai -3 juin 2022 paper from the perspective of manage and the manager and a strategical vision with a touch of human resources. We hope that our strategies will help pilot the human resources and the technostress difficulties in the companies. This paper is structured as follows. We begin with the literature review followed by the methodology, results, and conclusion.

LITERATURE REVIEW

COPING STRATEGIES FOR TECHNOSTRESS

Technostress

"In the last four decades, ICT has generated […] significant revenue for organizations in terms of reduced operating costs, process efficiency, new strategic alternatives, and opportunities for innovation" (Dos Santos & Sussman, 2000). ICT improves human and organizational performance [START_REF] Liang | Avoidance of information technology threats: A theoretical perspective[END_REF]. However, ICT is double-edged [START_REF] Lei | The double-edged nature of technostress on work performance: A research model and research agenda[END_REF]. When exploited for malicious purposes, ICT can pose a threat to individuals, organizations, and society. Organizations using ICT spend time and space avoiding these threats.

Technostress describes the stress that users experience because of application multitasking, constant connectivity, information overload, frequent system upgrades and [the] consequent uncertainty, continual relearning and [its] consequent job-related insecurities, and technical problems associated with the organizational use of ICT. [START_REF] Tarafdar | Impact of technostress on end-user satisfaction and performance[END_REF] Technostress creators such as techno-overload, techno-complexity, techno-invasion, technoinsecurity, and techno-uncertainty are associated with negative psychological and behavioral outcomes [START_REF] Pirkkalainen | Deliberate or instinctive? Proactive and reactive coping for technostress[END_REF]. [START_REF] Lazarus | Coping and adaptation[END_REF] stated that "Coping behaviors are those that an individual undertakes to tackle stressful situations." They thus defined coping as "an individual's cognitive and behavioral efforts exerted to manage specific external and internal demands that are appraised as taxing or exceeding the resources of the person."

COPING STRATEGIES FOR TECHNOSTRESS SITUATIONS

Existing research examined coping with work-related stress in work and family contexts [START_REF] Major | Coping with work-family conflict: A leader-member exchange perspective[END_REF]Michel et al., 2011), including heavy workload, organizational constraints, lack of autonomy at work, and interpersonal conflicts (Lu et al., 2010). [START_REF] Yip | Coping strategies as moderators in the relationship between role overload and burnout[END_REF] posit that "moderating coping strategies [affects] the relationship between role overload and burnout." Despite a serious lack of empirical data concerning the connection between specific coping mechanisms and defined stressful circumstances [START_REF] Dewe | Methods of coping with stress at work: A conceptual analysis and empirical study of measurement issues[END_REF], the coping strategies used to combat these specific stressors are not well presented in adaptation Annecy, 31 mai -3 juin 2022 scales in the literature on general stress [START_REF] Clark | Strategies for coping with work stressors and family stressors: Scale development and validation[END_REF]. We consider coping strategies for stress to find a general format that could apply in technostress contexts. [START_REF] Skinner | Searching for the structure of coping: A review and critique of category systems for classifying ways of coping[END_REF] support a highly personal, specific set of behaviors and strategies in stressful situations. They classify the coping process into lower and higher-level coping strategies. According to [START_REF] Pirkkalainen | Deliberate or instinctive? Proactive and reactive coping for technostress[END_REF], "The proactive coping behaviors of positive reinterpretation and Information and Technology (IT) control can help IT users by influencing the extent to which reactive coping behaviors are effective and by positively influencing ITenabled productivity." "Coping with the challenge [involves] actions and aims to achieve competence and mastery. They are activated when the individual experiences challenge"

[techno-stressors] [START_REF] Tarafdar | Explaining the link between technostress and technology addiction for social networking sites: A study of distraction as a Annecy, 31 mai -3 juin 2022 coping behavior[END_REF], and individuals respond to stressful or disturbing situations by engaging in coping behaviors [START_REF] Ragu-Nathan | The consequences of technostress for end users in organizations: Conceptual development and empirical validation[END_REF].

Coping strategies should be classified by type: emotion-focused and problem-focused coping. Coping researchers often distinguish between the two. Emotion-focused coping is usually defined as aiming to manage emotional distress associated with a situation (Lazarus & Folman, 1984cited in Baker & Berenbaum, 2007). Problem-focused coping involves efforts to solve a problem [START_REF] Carver | Assessing coping strategies: A theoretically based approach[END_REF][START_REF] Lazarus | Coping and adaptation[END_REF]. We explain the adaptive strategies of TMs in technostress situations via the coping dyad (interactional) approach.

MODEL AND PROPOSAL DEVELOPMENT

Figure 1 illustrates the process of interaction and reaction between TMs and employees. In the first phase, a deputy and subordinates interact in their positions for the first time. Regarding a new organization member, the first individual characteristic required special attention (Deinesch & Liden, 1986). Because such characteristics affect emotions in prior years, TMs can set up capacity testing with jobs corresponding subordinate positions [START_REF] Levinger | An 'incremental exchange' perspective on the pair relationship[END_REF] to compare and conduct periodic reallocation. Subordinates then answer to the delegation of a task, duty, or responsibility by the deputy (reaction or attribution). The last phase involves the core process of applying coping strategies to favorably resolve technostress. This interaction can be expressed by interdependence and LMX, which we discuss in the following section before presenting propositions by considering interdependence and LMX.

Participation and involvement

A participation and involvement strategy concerns teamwork, consultations with colleagues, collaboration, and exchange of information when conducting projects and applying different skills. We posit that positive interactions can curb technostress. Teamwork, coupled with collaboration and information exchange, help subordinates overcome challenges relating to technostress, especially when implementing new software or functionalities within the information system. However, if TMs and their subordinates work in isolation, they will be more prone to technostress than employees reporting challenges.

Proposal 1: Participatory interactions between TMs and subordinates lower technostress levels and improve team performance.

Mutual assistance

TMs assist their subordinates by sharing expertise and resources in case of a collaboration strategy. Mutual assistance is useful. Employees can help each other in matters related to information systems while facing challenges, which may increase productivity. TMs who refuse to assist with problems consider the technostress of subordinates as a competitive opportunity, but it provokes general stress and technostress. For example, subordinates will not alert the IT department or the hierarchy of the challenges they are facing, given the negative emotion induced by TMs, which will eventually increase technostress for the group Proposal 2: Collaborative interaction, mutual aid, and mutual assistance between TMs and subordinates lower technostress levels and improve team performance.

Procedure

The bureaucratic accommodation strategy explains that an initiative should not be procedural.

The author of an initiative has a proactive behavior and does not hesitate to identify or even Annecy, 31 mai -3 juin 2022 solve problems. For example, if a scanner does not work, the employee will use an alternative scanner without waiting for IT intervention. Conversely, when the procedure is followed to the letter, the subordinate only knows how to follow TM delegations passively. Thus, the slightest problem can become complicated and increase technostress levels, as no immediate solutions are sought.

Proposal 4: Bureaucratic interactions between TMs and subordinates lower technostress levels and improve team performance.

LMX

LMX theory views the dyadic relationship quality between leaders and members as the key to understanding effects of a leader on the members, teams, and organization [START_REF] Erdogan | Leader-member exchange (LMX) theory: The relational approach to leadership[END_REF]. However, TMs often ask firms to train a group of trustworthy employees to participate in their daily work, since they believe that time and effort are required to maintain a stable and trusting relationship with their employees. Often the question is whether a high-quality exchange can generate this type of relationship, which is characterized by a high degree of mutual trust, respect, and obligation [START_REF] Diaz | Communication technology: Pros and cons of constant connection to work[END_REF]. Conversely, poor quality exchange refers to a lack of trust, respect, and obligation [START_REF] Diaz | Communication technology: Pros and cons of constant connection to work[END_REF]. Therefore, the LMX approach is relevant here.

Given that LMX theory relies heavily on role theory [START_REF] Graen | A role-making model of leadership in formal organizations: A developmental approach[END_REF], roles are naturally linked to a set of expectations based on the prescriptions of others who interact with the system [or] organization in which they play a central role [START_REF] Harris | Resource loss from technology overload and its impact on work-family conflict: Can leaders help?[END_REF]. Roles are defined as standardized patterns of behavior required of all people who are involved in a given functional relationship [START_REF] Falconier | Dyadic coping and relationship satisfaction: A meta-analysis[END_REF].

LMX is built on a multidimensional foundation. We identify five dimensions, corresponding to the mutuality dimension between the manager and their subordinates: perceived contribution to the exchange, loyalty, affect [START_REF] Dienesch | Leader-member exchange model of leadership: A critique and further development[END_REF], professional respect [START_REF] Folkman | If it changes it must be a process: Study of emotion and coping during three stages of a college examination[END_REF], and trust (Bernerth et al., 2007). We explain the four proposals as follows.

Perceived contribution to the exchange

Perceived contribution to the exchange is the perception of the quantity, direction, quality of the work, and activity that each member puts forward for the mutual goals (explicit or implicit) of the dyad [START_REF] Dienesch | Leader-member exchange model of leadership: A critique and further development[END_REF]. Thus, managers examine the work performance of each subordinate on their team. Subordinates with better quality of work are appreciated and hold a more important place in the team [START_REF] Folkman | If it changes it must be a process: Study of emotion and coping during three stages of a college examination[END_REF]. This influences the quantity and Annecy, 31 mai -3 juin 2022 difficulty of the tasks delegated by TMs, who assign more challenging tasks to subordinates in whom they have the most confidence.

As the implementation of a work objective is extremely important and leadership orientation is a primary function, when TMs observe that subordinates have attained a quality of high contribution, they can communicate more information and provide employees with the necessary referral and support.

Proposal 5: Interactions focused on the perceived contribution to the exchange between TMs and subordinates lower technostress levels and improve team performance.

Loyalty

Loyalty strategy is the expression of public support for the objectives and the personal character of [other members] of the LMX dyad [START_REF] Dienesch | Leader-member exchange model of leadership: A critique and further development[END_REF]. TMs or subordinates promote objectives together, thus maintaining a good team spirit for a common benefit. TMs are more likely to entrust loyal subordinates with tasks that require independent judgement [or] responsibility [START_REF] Hollander | Leadership and social exchange processes[END_REF]. Moreover, loyal members can achieve higher levels of autonomy and job satisfaction, while less loyal members will have fewer responsibilities, less task autonomy, and low satisfaction levels. In summary, TMs associate loyalty and willingness with superior performance.

Proposal 6: Interactions based on the perceived loyalty between TMs and subordinates lower technostress levels and improve team performance.

Affect

Affective feelings should not be directed toward one thing, since a prototypical emotional element refers to a complex process that takes place over time, [and] involves causally linked sub-events [START_REF] Bass | Bass and Stogdill's handbook of leadership: Theory, research, and managerial applications[END_REF]. However, [START_REF] Dienesch | Leader-member exchange model of leadership: A critique and further development[END_REF] argue that the reciprocal affection of the members of the dyad is one over the other, based mainly on interpersonal attraction rather than on work or professional values. Thus, interactions between TMs and subordinates often occur when they need each other's company and are familiar with each other. Hence, the bonds of friendship often develop through professional interaction, friendliness is good for stress management and increases the cohesion of the working group.

Proposal 7: Interactions of affect between TMs and subordinates lower technostress levels and improve team performance.

Professional respect

Professional respect has been defined as the perception of the degree to which each member of the dyad has acquired a reputation, inside [and] outside the organization, for excelling in their Annecy, 31 mai -3 juin 2022 field of work [START_REF] Folkman | If it changes it must be a process: Study of emotion and coping during three stages of a college examination[END_REF]. Like affection, professional respect may appear as a factor that often has a negative connotation since professional respect is mostly related to power. For example, one who is perceived as professionally respected, would most likely be well connected within the organization and perhaps outside the organization due to importance in a profession (Ibarra, 1993). Thus, if people have good professional respectability, they can influence or generate results [START_REF] Russell | Core affect, prototypical emotional episodes, and other things called emotion: Dissecting the elephant[END_REF]. Accordingly, TMs and their subordinates can evaluate each other before working together or even meeting for the first time.

This consideration can be induced by reputation, expertise, and experience.

Proposal 8: Professional development interactions between TMs and subordinates lower technostress levels and improve team performance.

Mutual trust

Confidence is defined as a person's willingness to be vulnerable to another party whose behavior is not under their control based on the belief that the other person is competent, open, and reliable [START_REF] Bridge | Blended relationships: Friends as work associates[END_REF]. When TMs trust their employees, they give these employees preferential treatment. For example, they give them important information, more time, or rewards. Trust is based on cognition (e.g., knowledge of skills, reliability, or affect) [START_REF] Hosmer | Trust: The connecting link between organizational theory and philosophical ethics[END_REF].

Proposal 9: Interactions of mutual trust between TMs and subordinates lower technostress levels and improve team performance.

RESEARCH METHOD AND VALIDITY

RESEARCH METHOD

Exploring and testing are the two main processes of knowledge construction (Thietart, 2014).

Our research, therefore, fits well within the framework of a hybrid exploratory study because we seek an explanation of the adaptation strategies chosen by the MPs on technostress, which involves proceeding by going back and forth between empirical observation and theoretical knowledge (Thietart, 2014). We began with the dyadic adaptation theory by applying nine propositions, summarized through the notion of interdependence and LMX. The objective was to produce new knowledge.

VALIDITY

The concept of construct validity deals with one or more abstract concepts, which are not always directly observable [START_REF] Drucker-Godard | La gestion au quotidien des priorités du dirigeant [The day-to-day management of the leader's priorities[END_REF], and establishes correct operational measures for the concepts being studied [START_REF] Jung | Towards an ecological account of media choice: A case study on pluralistic reasoning while choosing email[END_REF]. Examples of such abstract Annecy, 31 mai -3 juin 2022 concepts include change, performance, power, relevance, and contribution. Thus, validity is abstract since it comprises abstract elements.

We evaluate the problem of technostress via technostress situations and adaptation strategies.

Hence, to guarantee the validity of the construct in qualitative research, appreciating the research objective and reviewing the literature is essential. This study aims to establish adaptation strategies under technostress, using interdependence and reciprocal relationship of leadership. We began with TM and subordinate interaction to determine the adaptation strategy to combat technostress. Thus, we present the three companies in our sample and the internal interactions regarding technostress, observe and describe the adaptation strategies, and analyze the adoption strategies.

Internal validity ensures the relevance and internal coherence of the results generated by the study [START_REF] Thiétart | Méthodes de recherche en management-4ème édition[END_REF]; however, we cannot identify a method for achieving a (good) level of internal validity of a search. To avoid internal non-validity, we followed [START_REF] Blome | Supply chain risk management in financial crises-A multiple case-study approach[END_REF] proposal to test rival hypotheses and compare the empirical diagrams highlighted in theoretical proposals from the literature. We conducted a literature review [START_REF] Ayyagari | Technostress: Technological antecedents and implications[END_REF][START_REF] Beaudry | Understanding user responses to information technology: A coping model of user adaptation[END_REF][START_REF] Katz | The social psychology of organizations[END_REF] to discover a possible link between technostress and the adoption of certain adaptation strategies.

To validate our study empirically, we adopted three phases. First, we collected data through semi-structured interviews. Then, we conducted a quality check by listening to the responses, followed by synthesis using keywords. We also requested ideas from the interlocutors. In the second step, we classified the responses by their main elements and translated them into proposals. A justification of the links between the TM adaptation strategy proposals and the nine proposals adopted makes it possible to identify the adaptation strategy. Third, we detailed the adaptation strategies focused on the type of adaptation (the problem or emotion).

DATA COLLECTION, ANALYSIS, AND RESULTS

Across three representative companies (Table 1), we interviewed 13 employees (Table 2). To convince different companies to accept our requests for interviews, we replaced the word technostress with the difficulty of using ICT and explained that the interviews would be anonymous and company names would not be mentioned. Table 3 presents the interview progress. We employed primary (observations and interviews) as well as secondary (internal documents and annual reports) sources of data. We also performed a pilot study using theoretical orientation [START_REF] Perry | Empirical studies of software engineering: A road map[END_REF] to obtain interviews. 10 Annecy, 31 mai -3 juin 2022

DATA COLLECTION

Company 2

This company operates in the services (insurance) sector. It is a European leader in insurance and has 7,200 employees and a turnover of more than 4 billion euros. The company was late in embracing ICT within its structure and policy. Nonetheless, the eventual introduction of different ICT tools was extensive. Within a few years, the company completely changed its information functions. This example is interesting because the company believes (rightly or wrongly) that the introduction of ICT and digitization of the claims management system will reduce the cost of processing files.

Company 3

The third company is a small and medium enterprise (SME), offering different e-learning services and learning applications on smartphones. It has 20 employees, and the entire workforce is constantly bathing in ICTs. Even if the employees seem very familiar with ICT, it is worth considering as an SME in this field. The study of this enterprise will be both general and comparative for the problems concerning our study.

We first collected and coded the interviews for each person and specified their positions and service. We then checked whether their responses were oriented toward a problem or an emotion, conformed with our initial proposal relating to whether people had access to more interdependent interactions or reciprocal relationships between TMs and subordinates, and whether TMs chose a problem-oriented strategy. Annecy, 31 mai -3 juin 2022

A classification of empirically collected adaptation strategies is explained in Section 4, where we explained the perspective of each interlocutor on their adaptive behavior associated with technostress. Therefore, to correspond to a hybrid study, we established a guide based on nine propositions via two main concepts -interdependence and LMX.

Regarding Company 1, the first step was to understand how executive managers view the behaviors of TMs for global technostress. Moreover, we acquired the feedback from employees on the difficulty of using ICT and took cognizance of the management difficulties encountered in the use of ICT, particularly the challenge of digitalization and digital information management. Here, executive managers learned the essential elements of our study and the main problems. They then gathered information and organized interviews. Company 2 recently experienced a change of head office and completed a digital transition. The first step was to understand the company through its organizational chart and mode of operation. We then conducted several interviews with managers from different departments; the average duration of these interviews was half an hour. In Company 3, we met manager Zhang, who explained to us the functions of the various departments of his company. Using the snowball method, after the interview, we requested for potential respondents for our questions.

We began our interview with a perspective to allow an open conversation on TM behavior and professional relationships. Following this, we compared the interviews, based on the nine propositions from the literature, with the propositions developed by us. Finally, the interlocutors revised their responses.

DATA ANALYSIS AND RESULTS

Data Analysis

Two data analysis phases were implemented to better analyze the results (Figure 2). The first phase classified the links between participants' responses and the proposals of the interactional coping strategies (ICS). Moreover, we translated them from keywords to coping strategy proposal focused on type. The second phase concerns identification of coping strategy.

Figure 2. Two phases of data analysis

Link between responses and proposals

Adaptation strategy proposal focused on type of adaptation Annecy, 31 mai -3 juin 2022

ADAPTATION STRATEGY PROPOSAL FOCUSED ON TYPE (FIRST PHASE)

According to witness responses on coping type, the following coping strategy items were identified by [START_REF] Carver | Assessing coping strategies: A theoretically based approach[END_REF]: problem-centered (active coping, planning, suppression of competing activities, coping to constraint, seeking instrumental social support) and emotioncentered coping (search for emotional, social support, positive reinterpretation, acceptance, denial, use of religion).

Proposal 1: Participatory interactions based on problem-centered coping.

Participatory interactions between TMs and subordinates are almost exclusively problemoriented, based on dialogue and communication addressing technostress. They focus on problem-solving, where the environment is participatory. Empirically, most adaptive behaviors of respondents are active coping (initiatives) (s2, s4, s5, s12, s13), planning (s1, s3, s8, s10), suppression of competing activities (s7, s11), and seeking social and instrumental support (s6, s14, s15). Each of these (s1, s2…etc.) corresponds to the numbering of the witness sentences in Table 4. Thus, there are participative interactions between TMs and subordinates, which result in lower technostress levels and improved team performance.

Proposal 2: Mutual assistance focused on the problem.

Collaborative interactions between TMs and subordinates are problem-oriented toward participatory coping (initiatives) (s17), planning (s18, s19), taking direct measures (s23, s24), requests for help (s16), seeking instrumental social support (s20, s22), and emotional (s16, s21)

reasons and obtaining moral support, sympathy, or understanding. Undoubtedly, there is mutual help, but it is often limited. It is linked to a trust that may not always exist between the various protagonists. It is, nonetheless, necessary that mutual help is used wisely; otherwise, it creates productivity problems. A good policy of mutual aid and assistance improves productivity and reduces technostress.

Proposal 3: Problem-centered conflict resolution.

Problem-oriented coping behaviors mainly emerge in this strategy: active coping (initiative; s23, s24), planning (s30, s31), taking direct measures (s25, s28, s29), deletion of competing activities (s7, s11, s27), and seeking instrumental social support (s26). Regarding the The bureaucratic interactions between TMs and subordinates are focused on the following problems: participatory coping (initiatives) (s30, s33, s37, s39), direct measurement (s37, s34), suppression of competing activities (s31, s33), and seeking instrumental social support (s35, s36). This strategy is fruitful on the condition that it is active, not passive, and is both upward and downward (superior to subordinate). Thus, if there are proactive exchanges and help, technostress can be reduced, and performance can be improved.

Proposal 5: Problem-centered perceived contribution.

This strategy applies to problem-oriented coping behaviors. It is strictly limited to adequate work behaviors (s41, s43) and involves acting or following guidelines in a given situation (technostress) (p. 42), collaboration (s39), and seeking social support from the manager (s43).

The more people contribute to their work within their teams, the more action they take toward addressing technostress. Hence, interactions based on the perceived contribution to the exchange between TMs and subordinates can reduce technostress and increase team performance.

Proposal 6: Emotion-based perceived loyalty.

Empirically, the type of associated emotions should directly influence coping strategies. This strategy relies solely on emotion, positive or negative. Positive emotion can enable optimistic focus on the technostress problem (s44, s45), as a positive reinterpretation means positive interpretation of appreciation or valorization by integrating and contributing within the team.

Thus, the higher the confidence, the greater is the emphasis the person places on supporting team members. However, negative emotion (s46) leads to harmful results, including failure to solve the technostress problem and decreased performance. Otherwise, dissatisfaction or frustration is generated, which leads to higher technostress levels and lower team performance.

Proposal 9: Problem-and emotion-based mutual trust Finally, mutual trust between TMs and subordinates is assumed to lower technostress levels and improve team performance. This strategy primarily focuses on the resolution of technostress by emotion. Thus, we can emphasize positive emotion (s57, s58, s59, s60, s61)

and the search for social support for instrumental reasons (s62). However, some respondents indicated that trust levels between colleagues are low. Table 4 presents the analysis of the various similarities and contrasts.

IDENTIFICATION ADAPTATION STRATEGY (SECOND PHASE)

Non-procedural strategy

Empowering employees allows them to adopt a participatory spirit and proactive initiatives, reducing hierarchical distance and developing a single communication channel between managers and subordinates. The following leads to the participative coping strategy (proposal 1):

Stéphanie speaks about the importance of participative management. According to her, if a technostress problem arises, she consults her manager to make a joint decision. Because in a stressful situation due to information overload for example, the best solutions go through communication, dialogue, support, and delegation. Julien explains that employees are convinced that the current project has good potential: "Above all, the manager has allowed us to subscribe to the company's capital" (for employees). This practice retains many of the employees. His colleague Li agrees with Julien: team spirit and a sense of creativity are important for a young company. Additionally, Aurélie emphasizes the importance of a sense of team: you must always stay connected and not isolate yourself from the group. Zhang Annecy, 31 mai -3 juin 2022 emphasizes their coping strategy to address technostress-they feel overwhelmed regardless of interventions.

Frédéric evokes team spirit and initiatives at work. According to him, there is a good team spirit among the colleagues, and the fact that they are able to digitize a document or to receive it already digitized facilitates its transfer to a colleague. Olivier explains that in his department, initiative and group work are important, he is rather familiar with ICT in this department since it is the heart of the profession, and he necessarily communicates with an ICT object apart from face-to-face verbal communication. In general, he approves of teamwork.

The sentiments necessarily follow the bureaucratic coping strategy (Proposal 4).

Frédéric thinks there are plenty of such initiatives. Olivier perceives that he collaborates with his superiors without a problem and that he shows initiative during his work. Similarly, Claude reveals that intervention or initiatives in this area to solve the problem can have a long-term perspective. That is, an intervention on his part or on the part of a colleague to help the other colleague who is in difficulty, and then if the problem is of a certain seriousness, one of them will take the initiative to talk about it with the superior.

Julien notes that sometimes the presence of an intermediary breaks the direct communication between employees and management. This seems to confirm Eric's answer: employees tend to force mutual assistance and participation, a type of mutual assistance and passive anticipation.

They show only a minimum interchange. Aurélie explains that her relationship with her supervisor involves exchanges and donations: "My supervisor and I have a two-way relationship. The voluntary actions on my part will be returned in one way or another by my manager. He is rather grateful for the initiatives I am taking" (Aurélie).

Professional support strategy

This strategy is supported by professional support, which means identifying problems, seeking solutions, implementing a policy of consensus and problem-solving, as well as participatory management. These are all methods used for the management or MPs and problem management of the information system in a harmonious way.

The situation brings us to the mutual assistance strategy (Proposal 2). "Is there mutual assistance between managers and subordinates in ICT?" Isabelle and Claude respond that "When there are difficulties, we try to help each other; it is mostly a question of time." Li and Wang, likewise, paint a friendly and anticipatory behavior between colleagues.

When it comes to things to do, conversations between my supervisor and me are generally The managing director of Company 3, Mr. Zhang, tells us about their adaptation strategy in fighting against technostress; he tells us that it is mainly based on the implementation of participatory management, entailing a policy of settling problems and inherent difficulties.

Their management always tries to cross the hierarchical distance to have a soft policy.

"Every time there is employee feedback, we give it a lot of importance, we put the issue (or employee feedback) on the agenda." (Zhang).

Achievement mechanism strategy

This strategy sets a clear compensation system in relation to individual and group contributions, gives importance to professional valuation, and evaluates employees more on their skills and expertise. Hence, it is vital to consider the contribution of a perceived exchange strategy (Proposal 5). Isabelle says her goal of helping or collaborating with her colleagues is based on a mutual team interest. Li mentions that each member must approach work adequately to enable work progression. Julien thinks his subordinates show more appreciation or cooperation.

Julianne says, "My manager told me that, thanks to me, the two groups are starting to communicate better; I am an outstanding member in this case."

With value comes the professional respect strategy (Proposal 8). With the introduction and evolution of ICTs in the company, Isabelle believes her work is valued. Éric feels he is valued; more concretely, this sense of valuation is based on bonuses. Claude does not think it is a good practice to be valued by financial compensation and believes that other valuation methods Annecy, 31 mai -3 juin 2022 should be preferred. Zhang says: For me, I value my employees through expertise.

Nevertheless, Stéphanie thinks ICT has helped her increase her contribution and productivity.

Mutual trust strategy

The manager and the team must establish a harmonious relationship of mutual trust, The strategy chain necessarily concludes with the mutual trust strategy (Proposal 9). Isabelle finds that the trust that already existed in the team remains intact. Frédéric confirms the opinions of Isabelle. However, Aurélie has enough confidence in her director; she is ready to defend him and justify his decisions even when he is absent. According to Zhang "I trust my employees because my employees are well selected. I believe in their skills and openness." Li says, "I respect the decisions and choices he makes about work." Isabel: there is mutual help (s16) Éric: helping one another (s16) Claude: everyone is ready to help the other (s16) Laurent: an initiative spirit has developed thanks to the introduction of ICT, as well as mutual trust (s17) Stéphanie: take stock of particular situations (s18), technical difficulties, sensitive topics, data that he cannot find, and what they lack (s19). They often share information, method, and achievements (s20) Li: there is a friendly atmosphere (s21) of mutual help, information sharing (s20), and anticipation between his colleagues (s3) Wang: My manager and I have sufficient conversations (s22) Yes Proposal 3: Conflict resolution and interactions between TMs and subordinates will lower technostress levels and improve team performance Isabel: understanding that beliefs, natural functions, and automatisms vary from person to person (s23) Olivier: able to differentiate between 'normal' problems and those caused by 'difficult' employees (s24) Frédéric: share with my colleagues and subordinate (s20) Li: A constructive and non-destructive attitude is paramount (s 25) Julianne: the importance of support and support for the individuals concerned in solving the problem of technostress (s 26) Zhang: it is necessary to avoid conflict and litigation (s27) Julien: put forward policies or assistance (p. 27) to help or support their subordinates (s16) Aurélie: software is well "disciplined" (s28) Wang: Once a new ICT is put in place or updated, it induces a corresponding training (by the manager) (s29)

Yes

Proposal 4: Bureaucratic interactions between TMs and subordinates will lower technostress stress Isabel: initiatives (s2); proactive (s30) Frédéric: good agreement (s31); thus, everyone takes the initiative (s2) Olivier: collaborates with his superiors (s32) Claude: reveals the intervention or initiatives (s2) Yes Annecy, 31 mai -3 juin 2022

CONCLUSION

Overall, there are four interactional coping strategies to fight against technostress: nonprocedural, professional support, valuation and contribution mechanism, and mutual trust strategy. Depending on the company, situation, and context, an interaction strategy can be better than others. We do not want to draw a catalog of strategies rather show managers and researchers the best strategy with a combination of elements (situation, company type, etc.).

Having a strategy is important; for example, when implementing a procurement strategy 4.0 through the moderating effect of information and the procurement strategy, there is a positive impact on the buyer intention [START_REF] Bag | Big data analytics as an operational excellence approach to enhance sustainable supply chain performance[END_REF]. The worst strategy is to not have a strategy, which is why we underline that the four strategies discussed are effective methods for coping with ICT.

THEORETICAL AND EMPIRICAL IMPLICATIONS

Theoretically, coping strategy classifications can enrich the technostress literature. We determined two other notions for adaptation strategy identification (interdependence and LMX). This interactional perspective broadens the field of research and helps us innovate the notion of coping, guiding TMs to improve emotional control and hence, resolve technostress problems.

Empirically, the problem of technostress cannot be denied. Moreover, the transformation of working methods and technological evolution generate new problems. Furthermore, there are different methods of adaptation and responses to technostress. However, not all of them result in the same outcome because adopting a response to technostress is not enough; the response must be adapted, else it would risk being counterproductive. The answers can be problembased or emotion-based. Furthermore, the proposals reduce technostress and improve team performance, despite operating at different levels. However, it is insufficient to merely apply Proposal 9: Interactions of mutual trust between TMs and subordinates will lower technostress levels and improve team performance Isabel: the previous team remained intact (s57) Frédéric: certainly, a limited trust (s58) remains between work colleagues. More generally, however, there is still confidence in how cases are addressed (s59) Laurent: I respect decisions and choices (s60) Zhang: I trust my manager (s49) because he was highly placed in a multinational organization; she believes in her skills and openness (s61) Aurélie: she is ready to defend and justify his decisions even when he is absent (s62)

Yes

Yes Annecy, 31 mai -3 juin 2022

Figure 1 .

 1 Figure 1. Interaction between TMs and subordinates

  manager and proximity manager, the intervention with respect to problemsolving and the emotions it provokes, is two-fold. Thus, the supervisor plays a vital role in the management of the difficulties related to technostress.Proposal 4: Problem-centered bureaucratic coping seeking instrumental social support(pp. 35-36).

Proposal 7 :

 7 Emotion-based affect.A perceived affect in the interactions between TMs and subordinates consists exclusively of emotion. Individuals try to manage the distress caused by the situation (technostress) rather than dealing directly with the problems. A positively oriented emotion instead, leads to a harmonized work environment and mutual trust (s47, s48, s49, s50,51). However, many Annecy, 31 mai -3 juin 2022 respondents did not mention the notion of affection. Thus, the professional environment is nonaffective. The relationship between TMs and subordinates remains somewhat neutral. Proposal 8: Problem-and emotion-based professional respect. This strategy focuses on the problem and emotion. Concerning the problem-oriented component, valuation by reward represents a search for instrumental social support (s52, s55, s56) and the suppression of competing activities (s53). Regarding the emotion-focused component, commendations from management correspond to social and emotional support (s54, s55, S56). Thus, giving high professional value reinforces employees' work commitment.

  effective. However, when work management has yet to set the course, faced with the mass of Annecy, 31 mai -3 juin 2022 information and software, you don't know where to start, my manager and I have the right number of conversations needed to get the job done. (Wang) Stéphanie mentions that when setting up or implementing new software, the teams often share information, methods, and achievements. In contrast, Éric perceives greater support between himself and his colleagues than with his superiors. However, despite this perception, he perceives the employees as isolated: "Each in his box in front of his computer." Thus, we must consider the conflict resolution strategy (Proposal 3). Aurélie maintains that their software is well disciplined and tamed by extensive use. Isabelle states "I try to evaluate the level of difficulties and the change due to new technologies." Olivier opts for a solution and considers differentiating the so-called normal problems and those caused by difficult information situations. Julianne indicates the importance of support activities. Zhang emphasizes it is necessary to make decisions and implement specific recommendations to avoid or manage conflicts and disputes.

  spirit (s1*) and initiatives at work (s2) Olivier: initiatives at work (s2) Julien: team spirit (s1) and participation in the project (s3) Li: the sense of creativity (s4), the possibility of subscribing to company capital for employees (s5) Aurélie: a sense of being part of the team (s1); you must always stay connected; do not isolate yourself from the rest of the group (s6) Wang: interprets the ideas of each other with precision between manager and subordinate (s7) Éric: for a common goal, they try to get along as best they can, help each other a little, and collaborate between colleagues (s8) Zhang: implementation of participative management (s9), problem-solving and difficulty policy (s10), and crossing the hierarchical distance (s11) Stéphanie: communication (p. 12), dialogue (s13), support (s14), and delegation

  the adaptation strategy, as encouraging certain methods of the chosen strategy is required, as in the case of valuation. This translates into a recognition demonstrated by the hierarchy surrounding the individual who can adapt and manage the transformation of the technological environment.We hope that this research deepens knowledge on how to cope with technologies. Learning the technostress coping strategies can help an individual enhance their technology experience, optimize the usage of technology, and derive more benefit from it. Finally, for further research, we can mobilize the resource-based view (RBV) based on the study of[START_REF] Bag | Big data analytics as an operational excellence approach to enhance sustainable supply chain performance[END_REF].

Table 1 . Company presentation

 1 This company is an industrial giant. It manufactures electricity products and offers services related to electricity management. It is a multinational company with markets in five continents; its main market is in France. It has 150,000 employees on its payroll and an annual turnover (excluding tax) exceeding 25 billion euros. Its operations extend to buildings, civil engineering, public works, and industrial and residential markets. The case is compelling as long as global communications and the mobilization of ICT tools by this company remain essential.

	NAME		COMPANY 1	COMPANY 2	COMPANY 3
	Activity area		Industrial	Insurance	Informatic service
	Workforce		+150,000 employees	+72,00 employees	Around 20 employees
	Annual turnover		25 billion euros +4 billion euros	/
	Company size		Large enterprise	Large enterprise	SME
	Type of market		International	National	International
	Geographical location		International	National	International
	Situation in ICT adoption	Leader in ICT adoption	Delay in ICT adoption policy	ICT adoption is an asset
	Common delimitation concerning technostress	We study coping strategies to technostress
				Table 2. Respondents' profiles
	RESPONDENTS		SERVICE
		Isabelle		Director of development affairs
	Company 1	Frédéric		Supply chain manager
		Olivier		Global public relationship service
		Éric		Assurance claims manager
	Company 2	Claude Laurent		Assurance claims manager in liabilities Assurance claims manager
		Stéphanie	Legal protection lawyer
		Zhang		General manager
		Julien		Informatic service
	Company 3	Li Juliana		Education service Administrative service
		Aurélie		Executive assistant
		Wang		Marketing service manager
				Table 3. Interview steps
	INTERVIEW PROGRESS	COMPANY 1	COMPANY 2	COMPANY 3
		Gain the perspective
	First step	of the manager on global coping strategies by managers on	Understand the organization chart and business operation	Global corporate presentation by its leader
		technostress

  cooperation, and respect and share a common goal. Surely, trust is closely associated with the loyalty strategy (Proposal 6). Julien mentions his superior considers his work with benevolence. Éric has a negative perspective: he thinks the relationship has not improved between subordinates and superiors. Wang states "I trust some managers and support them publicly; this trust develops in different contexts." Such affection invokes the affect strategy (Proposal 7). Li finds that his team share a friendly relationship. Aurélie states "I know the habit[and] the style of [our] director so well, my manager often says that I am a trustworthy person." Wang mentions "When I talk about how to do things at work, my supervisor and I are usually synchronized with each other. A habit of work is fixed between us, and it seems to be well."

TABLE 4 . PROPOSAL, WITNESS SENTENCE, AND COPING TYPE

 4

	*s1: witness sentence 1
	Proposal	Witness sentence	Coping type
			Focused	Focused
			on the	on the
			problem	emotion

ACKNOWLEDGEMENTS

We would like to thank the reviewers, the editors-in-chief, the guest editors, and the rest of the team for their reviews and comments. We would like also to thank the professors, researchers, and other members of AIMS. This paper is currently on press as a forthcoming paper in the IJTHI International Journal of Technology and Human Interaction (IGI global).

20

Annecy, 31 mai -3 juin 2022 and improve team performance.

Éric: he lacks initiative (s2) Julien: to exchange directly with the director (s33) Zhang: In societies that consider themselves free, many innovative aspects (s34) are regulated by spiritual freedom (s 35), open mind (s36), and autonomy (s37) Aurélie: My supervisor and I have a two-way exchange relationship (s33). Voluntary actions on my part are rendered in one way or another by my manager (s38) Proposal 5: Perceived contribution interactions to the exchange between TMs and subordinates will lower technostress levels and improve team performance Isabel: sharing a common interest (s1) to help or collaborate with colleagues or subordinate (s16) for a mutual team goal (s39) Julien: more appreciation or cooperation with my manager means he may give me more work and autonomy (s40) Li: Each member must approach the work adequately (s41) Zhang: Some people are delegated more stains or difficult spots or more quantity assignments depending on their ability, skill, and willingness (s42) Julianne: his manager knows his service contribution favorably (s43) Yes Proposal 6: Perceived loyalty interactions to the exchange between TMs and subordinates will lower technostress levels and improve team performance Julien: my manager admires him and regards him as someone faithful (s44) Julianne: he is personally inclined to use his power to help an employee solve work-related problems (s16) when he feels that the person is particularly faithful and shows a willingness to contribute to work (s45) Wang: I trust some managers and support them publicly; this trust develops under different contexts. (s46) Yes Proposal 7: Affect interactions between TMs and subordinates will lower technostress levels and improve team performance Li: his team shares a cordial relationship (s47); the members work together from the beginning of the project, and a mutual trust is created between the members (s48) Juliana: I can count on him to help me in case of difficulties (s49) Aurélie: I know the habits and style of the director so well (s50) Wang: My supervisor and I are usually synchronized with each other (s51) Yes Proposal 8: Professional respect interactions between TMs and subordinates will lower technostress levels and improve team performance Isabel: his work is valued (s52); certainly, there have been many changes in assignments Stéphanie: her work is not valued and is even less valued since the massive introduction of ICT (s53) Frédéric: my work is valued, but more concretely this valuation is based on premiums (s52) Éric: this valuation is based on premiums (s54) Claude: lack of valorization as rewards or congratulations received from management (s55) Zhang: I value my employees by expertise (s56)

Yes

Yes Annecy, 31 mai -3 juin 2022