Marie-Claire Saint-Lager
email: marie-claire.saint-lager@grenoble.cnrs.fr

Issam Laoufi

Aude Bailly

Operando atomic structure and active sites of TiO 2 (110)-supported gold nanoparticles during carbon monoxide oxidation

It is well known that gold nanoparticles supported on TiO 2 act as catalyst for the CO oxidation, even below room temperature. Despite extensive studies, the origin of this catalytic activity remains debated. Indeed, when the particle size decreases, many changes may occur; thus modifying the nanoparticles' electronic properties and consequently their catalytic performances. Thanks to a stateof-the-art home-developed setup, model catalysts can be prepared in ultra-high vacuum and their morphology are then studied in operando conditions by Grazing Incidence Small Angle X-ray Scattering, as well as their atomic structure by Grazing Incidence X-ray Diffraction as a function of their catalytic activity. We previously reported on the existence of a catalytic activity maximum observed for three-dimensional gold nanoparticles with a diameter of 2-3 nm and a height of 6-7 atomic planes. In the present work we correlate this size dependence of the catalytic activity to the nanoparticles' atomic structure. We show that even when their size decreases below the optimum diameter, the gold nanoparticles keep the face-centered cubic structure characteristic of bulk gold. Nevertheless, for these smallest nanoparticles, the lattice parameter presents anisotropic strains with a larger contraction in the direction perpendicular to the surface. Moreover a careful analysis of the atomic-scale morphology around the catalytic activity maximum tends to evidence the role of sites with a specific geometry at the interface between the nanoparticles and the substrate. This argues for models where atoms at the interface periphery act as catalytically active sites for the carbon monoxide oxidation.

Introduction

It is now well established that gold nanoparticles (GNPs) supported on rutile TiO 2 act as catalysts for the oxidation of carbon monoxide (CO), even below room temperature. 1 that case the active sites are mainly identified as low-coordinated gold atoms. 3 The GNPs catalytic activity can also be due to quantum size effect linked to a metal-non metal transition appearing for clusters with thickness smaller than 1 nm. 4 Haruta et al. first suggested that the reaction could take place at the interfacial perimeter between the GNPs and the substrate. 5 This assumption was recently supported by studies that stress the crucial role of this perimeter in the CO oxidation, but the reaction mechanism is still strongly debated.

Kotobuki et al. demonstrated with temporal analysis of reactive products that active oxygen species can be stored on Au/TiO 2 catalysts at 350 K by exposure to thermal O 2 pulses. 6 They showed that the oxygen storage capacity and the CO oxidation activity of these catalysts scale with the perimeter length of the interface between the TiO 2 support and the GNPs. In the continuity of this work, Widmann et al. proposed a mechanism where CO adsorbed on the GNPs reacts with activated surface lattice oxygen species at the perimeter of the Au-TiO 2 interface. 7 At room temperature only the perimeter sites are involved whereas at higher temperatures (> 350 K), these oxygen species can be provided by migration on TiO 2 via oxygen vacancies. The authors concluded that those highly stable active oxygen species allow the facile dissociation of the oxygen molecule without assistance of any intermediate complex.

By means of a combination of infrared kinetic spectroscopic measurements and Density Functional Theory (DFT) calculations, Green et al. evidences that dual Ti-Au catalytic sites located at the perimeter interface of 3-nm GNPs supported on TiO 2 play a key role in the CO oxidation. 8 They indicate that O-O bond scission is activated by the formation of a CO-O 2 complex at these dual sites. They also stress the possibility for CO to adsorb either on Au or on TiO 2 sites thus giving two different sources of CO for the oxidation reaction, whereas the oxygen activation takes place around the perimeter area. These experiments carried out at low temperature (about 110 K) make it possible to distinguish which CO adsorption sites will first deliver the CO for the reaction indicating that the TiO 2 sites are energetically more favorable at such a low temperature. The CO adsorbed on top of the GNPs will only be able to diffuse to the active perimeter area at higher temperatures.

Fujitani et al. also stressed the importance of the nanoparticle-support periphery. 9 But according to the fact that CO oxidation on supported GNPs is greatly influenced by moisture, they suggest that reaction intermediates are rather OH-species.

A lot of attention was also dedicated to visualize GNPs at the atomic scale in real space and under reactive conditions. Yoshida et al. showed that the adsorption of carbon monoxide at room temperature induces the reconstruction of {100} facets of GNPs supported on CeO 2 during the CO oxidation. 10 The aberration-corrected environmental Transmission Electron Microscopy (TEM) images evidence that the carbon monoxide is preferentially adsorbed on the on-top sites of the gold atoms; the outermost gold layer forming an undulating hexagonal lattice. This reconstruction is observed for total pressures as high as 2.10 3 Pa. Although CO was thought to preferentially adsorb on the low-coordination sites, such as steps, edges and corners, this study evidences that atomically clean reconstructed {100} facets of roughly 4 nm-diameter GNPs are also able to adsorb it. This illustrates that gaseous environments can strongly modified the GNPs atomic structure at the nanoscale. But deep structural changes can occur even in Ultra-High Vacuum (UHV) thus modifying the nanoparticles' electronic properties and consequently their catalytic properties. By means of aberration-corrected TEM, Walsh et al. studied carbon-supported gold nanoparticles in the size range for which they are usually known to be catalytically active, i.e. 2 to 10 nm. In UHV the multiply twinned icosahedral and decahedral structures turn out to be the most stable ones. 11 These highly strained GNPs (especially at the surface) are expected to be much more active than the single crystalline ones. Indeed, based on powder X-ray diffraction, Cunningham et al. found that, for Au/Mg(OH) 2 , only catalysts with size distributions below 1 nm were active for CO oxidation. They identified two structural forms (cubo-octahedral and icosahedral) and observed that only the icosahedral one is catalytically active. 12 However such strained structure with a great number of lowcoordinated atoms at the surface does not seem to be stable under annealing or even under the microscope electron beam inducing sintering with the appearance of atomically clean surface facets in order to minimize the surface energy. 13 Anyway a lattice contraction is generally observed in very small particles. A decrease of the Au-Au distance with decreasing GNP size below a few nanometers was measured by Extended X-ray Absorption Fine Structure (EXAFS). According to the authors this effect could be responsible for the catalytic properties of the gold nanoparticles. 14 This brief overview attests that all the recent developments allowing the observation of the structure at the atomic scale in reactive conditions pave the way toward a thorough understanding of the catalytic properties of supported GNPs. The present work comes within this scope. Thanks to a state-of-the-art home-developed setup, model catalysts can be prepared in UHV and their morphology then studied in operando conditions by Grazing Incidence Small Angle X-ray Scattering (GISAXS), as well as their atomic structure by Grazing Incidence X-ray Diffraction (GIXRD) as a function of their catalytic activity. 15,16 It was used to characterize gold nanoparticles supported on TiO 2 (110) during CO oxidation, by recording at the same time the catalytic activity, GISAXS patterns and GIXRD data. We previously reported that the reaction rate dependence on the GNPs diameter presents a maximum for diameters of about 2 nm. 17,18 Above the maximum, the reaction rate well fits a power law of the diameter, 20 that is consistent with low-coordinated active sites located at the GNP interface periphery. However, this cannot explain the activity fall observed for GNPs smaller than 2 nm. The analysis of the evolution of the GNPs' atomic-scale structure and morphology around this maximum should yield a better understanding of the key parameters involved in their catalytic properties. Moreover, in contrast to atomic structures determined by X-ray measurements performed on powders, the use of nanoparticles epitaxially grown on a TiO 2 (110) single crystal allows to get information on their shape, their atomic structure and their lattice parameter parallel and perpendicular to the substrate surface. In this frame, GISAXS and GIXRD lead to define what can be called an 'average' particle analyzed in the reciprocal space and representing a wide collection of nanoparticles. In that sense it is a good complement of local probes such as TEM or Scanning Tunneling Microscopy (STM). Moreover these X-ray techniques can easily be operated in operando conditions, the simultaneous determination of the reaction rate allowing a direct correlation with the geometrical parameters of the average GNP.

Experimental method

The experiments were performed in a home-developed setup consisting of an UHV preparation chamber connected to a batch reactor running from UHV up to reactive conditions at ambient pressures and allowing both X-ray scattering and catalytic activity measurements in static conditions. 18,19 The setup is operated on the "2+2" diffractometer of the GMT station at the beamline BM32 of the European Synchrotron Radiation Facility (ESRF). 17 The experimental procedure was previously detailed; 19,20 we summarize the main features hereafter. The GNPs were synthesized in the UHV preparation chamber (base pressure of 5.10 -8 Pa). The TiO 2 (110) single crystal, used as a substrate, was first bombarded with 750 eV argon ions and then annealed at 1000 K under an oxygen partial pressure of 10 -3 Pa in order to restore the surface stoichiometry. 19 Gold nanoparticles were grown by UHV vapor deposition at 300 K from a Knudsen cell. As calibrated by a quartz microbalance, the average thickness of the gold deposit was ranging between 0.05 and 3 atomic monolayers (1 ML corresponding to 0.235 nm, equivalent to one Au(111) atomic layer). Once prepared, the Au/TiO 2 (110) samples were transferred under UHV into the batch reactor in order to perform X-ray and catalytic activity measurements during the carbon monoxide oxidation. The measurement procedure involved three steps; each lasting approximately two to three hours: (1) Au/TiO 2 (110) was first studied in UHV (2.10 -7 Pa) at room temperature (RT), (2) then during annealing at 473 K in 2.10 3 Pa of oxygen, (3) and finally during CO conversion into CO 2 that started by adding 20 Pa of CO to the oxygen, while keeping the sample at 473 K. The CO and CO 2 partial pressures in the reactor were deduced respectively from the 28 and 44 ionization currents measured by a mass spectrometer. The gas composition was followed as soon as the sample was introduced in the batch reactor and all along the experimental procedure. During the step (3), the reaction rate was defined as the number of CO molecules converted into CO 2 per gold atom per second on the basis of the total number of gold atoms. 19 The GISAXS patterns were generally acquired at the beginning and at the end of each step. In between, diffraction spectra were collected along several directions in the reciprocal space.

To enhance the GNPs' signal, X-ray measurements were performed with a photon energy of 18 keV at a grazing incidence close to the critical angle (0.13° for TiO 2 at 18 keV). A shutter, opened only during X-ray data collection, was mounted in order to avoid long time exposures to the X-ray beam and consecutive sample surface damages. Finally, no significant difference could be observed in either the scattering or the catalytic activity with and without photons on the sample. For GIXRD measurements, a standard NaI scintillator detector was used and the GISAXS patterns were collected on a 2D camera (Princeton Instruments, pixel size of 56.25 μm). 18 The structure of bulk gold is face centered cubic (fcc) and two kinds of epitaxy relationships are usually observed on TiO 2 (110) : (112) Au //(110) TiO2 and (111) Au //(110) TiO2 , depending on the substrate temperature during the GNPs growth. 20 However, by means of GIXRD in UHV, Lazzari et al. found that these two epitaxial relationships are nearly equiproportional whatever the temperature. 21 For the two epitaxy orientations gold grows with the <110> Au direction of the dense rows aligned with the <001> TiO2 of the oxygen ones, the interatomic distances being 0.296 nm for TiO 2 and 0.2886 nm for gold. As usual in surface crystallography, the z direction is perpendicular to the surface, that means along the <110> TiO2 direction, and the x direction has the smallest lattice parameter that is along <001> TiO2 . The unit cell parameters are thus equal to a = 0.296 nm, b = c = 0.6492 nm. The present diffraction measurements show that the (111) Au //(110) TiO2 epitaxy is predominant, especially under reactive conditions. For convenience the gold lattice is described in its hexagonal cell. The schematic representation of the fcc gold reciprocal space (with Miller indices: H Au , K Au , L Au ) over the TiO 2 (110) one (with Miller indices: H TiO2 , K TiO2 , L TiO2 ) is given in Fig. 1a for L = 0. Fig. 1b is a 3D schematic representation of the gold reciprocal space for the two kinds of stacking usually observed along the <111> trigonal axis.

For each gold deposit, the mean geometrical parameters of the GNPs, such as the diameter D, the height H and the interparticle distance L, were deduced from the quantitative analysis of the GISAXS patterns performed using the IsGISAXS software. 22 These measurements are detailed elsewhere. 18,19,20 Considering the whole set of samples measured in operando, the best fit of the GISAXS patterns was obtained assuming a truncated sphere to model the GNPs shape. The geometrical parameters are an average value of those obtained for the two main TiO 2 (110) surface directions. In contrast to the interparticle distance L, the diameter D was found identical in these two directions with decreasing value when the deposited equivalent gold thickness diminishes. Fig. 2 sums up the main results concerning the dependence of the CO conversion rate as a function of the mean geometrical parameters deduced from the quantitative analysis of the GISAXS patterns recorded in real time. 19

Results

Evolution of the diffraction spectra under reactive gases

The evolution of the diffraction spectra obtained during the three steps of the experimental procedure is illustrated in Fig. 3 for a sample with an equivalent gold thickness of 0.1 ML (Fig. 3a and3c) and another one with 0.5 ML (Fig. 3b and3d). Two directions of the reciprocal space are represented: parallel to the surface along H and perpendicularly to it along the rod (0 1) Au .

For the H-scan, the Au (2 -1 0) Au Bragg peak is expected at H TiO2 = 2.05 (Fig. 3a and3b). In UHV, its position is found at a higher value, and the shift is larger when the gold thickness decreases (corresponding to smaller particles). For 0.1 ML, the quantitative GISAXS analysis gave 1 nm for the diameter and 0.6 nm for the height and the Au (2 -1 0) Au peak was found at H TiO2 = 2.11 ± 0.02. It corresponds to a lattice contraction of 3.4 %. In the case of the 0.5 ML deposit, the nanoparticles are larger; the diameter being 2.2 nm and the height 1.5 nm. The gold maximum is found at H TiO2 = 2.07 ± 0.01 indicating that there is still a small lattice contraction of 0.8 %. When annealing at 473 K and also after introducing CO for starting the oxidation reaction, strong changes are observed for the thinnest deposit, whereas at 0.5 ML there is only a gentle evolution. Above 1 ML, the spectra remained independent of the environment. This behavior can be correlated to what was deduced from GISAXS patterns measured at the same time.

Under reaction conditions (step 3), for the 0.1 ML sample, the diameter grows up to 1.7 nm and the height to 1.2 nm, as deduced from GISAXS, and the gold peak shifts to H TiO2 = 2.06 ± 0.02, a value closer to the bulk gold one. For the 0.5 ML deposit, the peak reaches the position of the bulk value. In that case, we can observe that there is a subtle difference between step (2) and step (3); the relaxation seeming slightly larger under oxygen than under reactive conditions.

Along the (0 1) Au rod and during the step (1), in UHV and at room temperature, the intensity collected for 0.1 ML, with a height of about 3 atomic planes, presents only one main broad peak (Fig. 3c). For the 0.5 ML, the height corresponds to at least 6 atomic planes and in that case, the diffraction spectra exhibit well defined peaks at L Au = 1, 2, 4, 5 (Fig. 3d) characteristic of the two sequences ABC and ACB of the fcc stacking. In the case of the 0.1 ML deposit, the fcc ordering is progressively induced by gas adsorption at 473 K. Indeed, under oxygen at 473 K (step 2), the main broad peak starts to split into two peaks. Then, under reaction conditions at 473 K, the nanoparticles height increases up to 6 atomic planes. In the same conditions, the peaks of the 0.5 ML deposit only slightly increase and become a little bit thinner.

Au-Au distances and catalytic activity

The interatomic distances d Au-Au in the gold nanoparticles during reaction were deduced from the diffraction spectra. Perpendicularly to the surface, it was obtained by averaging the position of the peak at L = 2 and 4 along the (0 1) Au rod with the one for L = 3 of the (2 -1) Au rod. This choice allowed cancelling the shift of the peaks due to defects in the fcc stacking. In the surface plane, it was calculated by deconvoluting the gold peaks from the TiO 2 one in the radial scan around (200) TiO2 . Besides, we verified that the width of the gold peaks is consistent with the size deduced by GISAXS. The variation of the Au-Au distances is reported in Fig. 4 as a function of the particles diameter. They were normalized to the bulk gold values that are d ⊥ = a 0 /√3 and d // = a 0 /√2; the bulk Au cubic lattice parameter a 0 being 0.408 nm. A contraction is observed for sizes below ≈ 2-3 nm, as already reported by using EXAFS. 16 But in the present case, the Fig. 4 shows that this effect is clearly anisotropic: it is almost negligible in the surface plane whereas it is quite significant perpendicularly to it. This can be correlated to the shape of the nanoparticles, since the aspect ratio H/D is less than 1 and decreases in this region from 0.7 to 0.5. Interestingly, the comparison with the variation of the catalytic activity for CO oxidation in Fig. 4 indicates that this contraction coincides with the fall of the catalytic activity below the optimum diameter.

Geometry of the active sites

Looking at the geometrical parameters of the nanoparticles corresponding to the drop of the activity below the optimum diameter, we observed that the slope of the variation of the mean height H as a function of the diameter becomes more abrupt (Fig. 2). This means that nanoparticles become flatter in this particular region. Actually, in the region around the maximum of activity, the value of the mean diameter varies slowly while the height changes more significantly. It is thus interesting to plot the variation of the GNP geometrical parameters as a function of the particles height and to correlate it to that of the activity for CO oxidation such as in Fig. 5. It shows that the diameter is more or less constant in the region of the maximum of activity indicating that it is not the essential parameter. At first sight, the models, where the reaction occurs at the interface perimeter between the GNPs and the TiO 2 surface, would expect that the variation of the CO conversion rate per Au atom would follow the same variation as the fraction of Au atoms on the perimeter. 11 Fig. 5 shows that it is not the case here since this fraction continuously increases when the particles size decreases, including while crossing the maximum of activity. In contrast, the aspect ratio H/D presents a behavior similar to the activity with a maximum for the same value of the height indicating that the geometry of the particles at the interface is a crucial parameter. The schemes inserted in fig. 5 evidence that the maximum of activity corresponds to a sharp angle θ between the GNP surface tangent and the substrate. The fall on both sides of the maximum corresponds to θ becoming closer to a right angle. This suggests that the active sites could result from the combined action of several atoms near the perimeter. Thereby, a kind of cavity is created where low-coordinated Au atoms and the titania substrate can interact simultaneously with the reaction entities.

To get more informations we need to use an atomic model to describe the GNPs structure. Most of the images obtained by TEM with atomic resolution evidenced a shape close to cubo-octahedrons for particles size in the range of the present work and moreover with a (111) face in contact with the substrate surface. 23,24,25 Consistently, as shown hereabove, the diffraction data collected for sample under reactions conditions reveal a fcc structure with the <111> direction perpendicular to the surface that is compatible with such cubo-octahedrons. 26 However, we did not find any trace of signal indicating well-defined facets on the recorded GISAXS patterns, which would present oriented rods corresponding to the (111) and (100) planes as observed for well-ordered facetted supported clusters. 27 This may be due to a misorientation of the gold nanoparticles or to particles too small to present facets large enough to be seen by GISAXS. Therefore the truncated sphere used to model the nanoparticles for quantitative analysis of the GISAXS patterns is a quite good approximation. The two parameters needed to define the corresponding truncated cubo-octaedrons are m and l, the number of atoms on the edges and the number of atomic planes parallel to the surface, respectively. Their relationships with the diameter D and the height H measured by GISAXS were established in ref . 20 As shown in Fig. 6, the experimental data do not exactly match with integer values of m and l. They represent the center of the size distribution of the GNPs on the TiO 2 substrate. The width of this distribution in the two directions is represented by error bars in the insert of Fig. 6. For the height, it can be interpreted as a distribution of particles with different number of atomic layers; it varies noticeably as a function of the particle size. For the flattest nanoparticles, the error bar is large and due to a wide distribution of the size on the sample. For the biggest ones, the truncated sphere becomes a less good approximation of the GNPs shape that begins to present well defined facets, enlarging the value of the error. In the middle range, that means close to the optimum size for the activity, the particles height has a thin distribution (l = 7 ± 0.5 at the maximum). As previously reported this is a consequence of the sintering induced by the temperature and then by the reaction itself. 19,20 Concerning the dispersion width of the diameter, it is roughly constant and corresponds to ± 2 atoms. Thus, it cannot be attributed to a distribution of octahedrons with several m integers. Indeed, a distribution with m and m + 1 atoms on the edge would correspond to diameter with N D and N D + 3 respectively and would thus result in a bimodal distribution for the diameter, that is definitively not observed. This indicates that GNPs shape is not ideal. For instance, a twin structure was evidenced for nanoparticles exposed to 1 vol % CO in air (100 Pa) by High Resolution TEM (HRTEM). SOM27 However, we can observe that all the experimental data are localized close to an integer value of m (less than ±1 atom), that means with a shape not far from the ideal cubo-octahedron model. Fig. 6 shows that the data below the optimum have m close to 3, when the number of layers l grows, the reaction rate increases (vertical red arrow in Fig. 6). Above the optimum size for the (almost) same number of layers, the activity increases when m goes down to 3 (horizontal red arrow in Fig. 6). For higher values of m, the main factor prevailing for the activity fall is the rapid decrease of the fraction of potentially active sites at the surface relatively to the total number of atoms in the particles. Finally, the maximum of activity occurs for particles close to cubo-octahedron with m = 3. The main feature of this GNPs morphology is the presence of edges parallel and close to the substrate surface. They connect a (111) hexagonal face of the upper part of the particles and a (100) square face that makes a sharp angle of 55° with the TiO 2 surface. The combination of such an angle, together with low-coordinated gold atoms on the edges in the vicinity of the substrate surface constitute a particular geometry that could be very active sites for the CO oxidation.

Conclusion

In this work we studied the structural properties and the morphology of the GNPs around the maximum observed for the CO oxidation reaction rate as a function of the particles diameter, thanks to in operando GISAXS, GIXRD and activity measurements. For GNPs bigger than 2-3 nm, that is above the optimum size for the catalytic activity, the fcc structure is already established during the gold growth on TiO 2 (110). For the smallest ones, the fcc ordering appears only after annealing at 473K and under reactive gas pressure. Before adding 20 Pa of CO, starting the oxidation reaction, the GNPs were exposed to 2.10 3 Pa of oxygen at 473K. The diffraction data present a subtle difference between the two steps with the relaxation of the Au-Au distance that seems slightly larger under oxygen than under reactive conditions. This may be due to oxygen interaction with outermost atoms of the GNPs which could provoke a dilatation of the Au-Au distances (the measured value is an average over the particle). As already reported, under oxygen pressure, gold is probably pre-oxidized, thus producing Au δ+ sites on the GNP surface. 29 Such positively charged Au site was also evidenced under high pressure of oxygen at room temperature by X-ray Photoelectron Spectroscopy (XPS), 30,31 as well as by X-ray Absorption Near-Edge Structure (XANES). 32 This gold activation was attributed to Xrays. 33 Such a pre-oxidation step was shown to modify the reaction pathway at low temperature (120K), since it deactivates the most reactive sites. 31 However, in most of the cases, CO introduction quickly reduced gold to its metallic form, 33,34 all the more since the temperature is equal to 473K in the present study. The interaction of oxygen with GNPs was also observed by HRTEM leading to rounded GNPs. 33 In a more recent work, Uchiyama et al. drew the diagram of the GNPs morphology grown on CeO 2 as a function of the O 2 and CO partial pressures. 26 According to it, the pressure conditions of step (2) of the present study correspond to the domain where the GNPs are rounded. In reaction conditions (i.e. after CO introduction) the diagram indicates that the GNPs should be well facetted.

Under reaction conditions, the GNPs have the fcc bulk structure with the <111> Au direction perpendicular to the TiO 2 (110) surface. However, for diameters below 2-3 nm, they are anisotropically strained with a contraction perpendicularly to the surface that can reach 2% whereas it is negligible in the parallel direction. This contraction of d Au-Au coincides with the fall in the catalytic activity below the optimum size. A deeper structural analysis is in progress to determine if more significant structural changes occur in this size range to explain the fall of the GNPs catalytic activity.

In the present work, we show that the maximum of activity corresponds to a sharp angle between the GNP surface tangent and the substrate. This work confirms the dominating role of the sites at the interface but with a specific geometry involving substrate atoms and several atoms of the GNPs. The temperature being 473 K, the diffusion on the GNPs surface as well as on the TiO 2 surface cannot be the limiting factor as for studies done at much lower temperatures (around 100 K). 9,31 This would claim for models involving the existence of intermediate complexes on mixed adsorption sites to activate the oxygen molecules.

These results on TiO 2 (110)-supported GNPs show that the combination of GISAXS, GIXRD and catalytic activity measurements is a very promising way to examine at the atomic scale nanoparticles with sizes around 2 nm and below while they are catalyzing a given reaction.

Figures captions

Fig. 1: Schematic representations of the reciprocal space of (111) Au //(110) TiO2 (a) at L=0 with in black the TiO 2 peaks (filled circles: Bragg peaks and empty circles: cut of the surface rods) and in orange the gold Bragg peaks. H TiO2 , K TiO2 and L TiO2 are the Miller indices in the frame of the TiO 2 (110) surface. The hexagonal cell of the fcc bulk gold is represented in orange. (b) 3D representation of the gold reciprocal space. The orange spots represent the Bragg peaks of the ABC stacking and the blue ones those of the ACB one. The dark red spots are peaks common to the two stackings. In the two schemes the green arrows indicate the directions along which the diffracted intensity was collected for Fig. 3. Au atom per second, (blue) diameter D of the GNPs, (grey) ratio between N P and N V , the number of atoms at the interface perimeter between the nanoparticles and the TiO 2 substrate and the number of total number of atoms within the GNPs, respectively, and (red) the aspect ratio H/D. A schematic representation of the supported nanoparticles is given with, on the left, H/D = 0.5 and on the right, H/D = 0.7 as observed close to the catalytic activity maximum. θ is the angle between the GNP surface tangent and the substrate.

Fig. 6:

Reaction rate of the GNPs as a function of the size of the corresponding truncated cubooctahedron with, on the vertical axis the number of atomic layers l, and on the bottom horizontal axis, m, the number of atoms on the edges and on the upper one, N D , the number of atoms along the diameter; the scales for l, N D and m are proportional to the value of d ⊥ , d // and 3d // , respectively. The red spots are proportional to the reaction rate for CO oxidation, as deduced from the experimental data. They are located at the m and l values deduced from the mean D and H, as explained in the text. Schematic representations of cubo-octahedrons are given for several couples of (m,l) pointed by the blue arrows (see text for the red arrows). The couples (m,l) corresponding to complete cubooctahedron are marked by dark blue diamonds. On the lower grey dashed line, the aspect ratio H/D is 0.5 and it is 0.8 on the upper one, that crosses the dark blue diamonds. The part of the diagram above it has been shaded since the corresponding (m,l) couples cannot be assigned to the cubo-octahedral geometry. In the insert, the experimental data were also plotted as a function of l (vertical axis) and N D (upper horizontal axis) with their error bars calculated from the width of the Gaussian distribution of the GNPs size given by the GISAXS analysis.

in line with reference: 28 3m - 2 ≅

 282 N D = (D -2r Au )/(d // + 1) and l = (H -2r Au )/(d ⊥ + 1), N D being the number of atoms along the diameter D and r Au = 0.026 nm, the gold atomic radius as seen by X-rays. The variation of the interatomic distance d Au-Au mentioned above is neglected here and the bulk values for d ⊥ and d // are used.

Fig. 2 :

 2 Fig. 2: Variation of the CO conversion reaction rate into CO 2 per Au atom per second (black) and of the mean height H (blue) as a function of the mean diameter D (blue) of the GNPs measured in operando. The full lines are guides for the eyes.

Fig. 3 :

 3 Fig. 3: Diffracted intensity: (a) and (b) along a radial scan around (2 -1 0) Au and (c) and (d) along the (01) Au rod for two samples with gold equivalent thicknesses of : (a) and (c) 0.1 ML and (b) and (d) 0.5ML during the three steps of the experimental procedure. (1) in UHV at RT (black), (2) under 2.10 3 Pa of oxygen at 473 K (blue) and (3) after adding 20 Pa of CO while keeping T = 473 K (red). In (a) et (c) the thin peak centered at (200) TiO2 with a broad and intense feet is the TiO 2 Bragg peak, the shoulder on the right is the (2 -1 0) Au Bragg peak that is expected at (2.05 0 0) TiO2 for bulk Au as indicated by the grey dashed line.

Fig. 4 :

 4 Fig. 4: Variation of the CO conversion reaction rate into CO 2 per Au atom per second (black) and of the interatomic distance d Au-Au , normalized to the bulk gold value, (blue and red : perpendicular and parallel to the surface, respectively) as a function of the GNPs' mean diameter D. All the parameters were measured in operando during CO oxidation (step (3)): the catalytic activity by mass spectrometry, the diameter D was deduced from GISAXS and d Au-Au from the diffraction measurements. The full lines are guides for the eyes.

Fig. 5 :

 5 Fig. 5:Comparison of the variation of the GNPs geometrical parameters around the catalytic activity maximum as a function of the height. (black) Reaction rate of the CO conversion into CO 2 per Au atom per second, (blue) diameter D of the GNPs, (grey) ratio between N P and N V , the number of atoms at the interface perimeter between the nanoparticles and the TiO 2 substrate and the number of total number of atoms within the GNPs, respectively, and (red) the aspect ratio H/D. A schematic representation of the supported nanoparticles is given with, on the left, H/D = 0.5 and on the right,

  Figure 1

Acknowledgements

The technical staffs (SERAS and MCMF) of Institut Neel-CNRS and of the BM32 CRG beamline at ESRF are acknowledged for their assistance. We also thank Rémi Lazzari and Jacques Jupille of the INSP for their help during the synchrotron experiments."


Maks Banens

some-observations-oneuropean-diversity

Same-sex unions gained considerable legal recognition in Europe over the last few decades. Denmark (1989), Norway (1993), Sweden (1995) and Iceland (1996) were the first countries that introduced the so-alled i il pa t e ship fo sa e-sex couples. These partnerships offered somebut not allrights and duties generally associated with marriage. The provisions that were primarily excluded from these statutes included shared parenthood and the o d a iage . Othe West-European countries followed and soon same-sex marriage, including shared parenthood recognition, was legalized. This major transformation spread over Western Europe, from North to South, with some countries joining the general movement quicker than others. From a distance, the changes looked relatively uniform.

 found that European same-sex union frequencies were not particularly low and concluded that couples did register as soon as they had any interest in doing so. Many scholars joined either side of the debate and discussed normalization, hetero-normativity, gay citizenship and more generally the social organization of sexuality through gay marriage. My research does not argue for or against same-sex union registration. It uses same-sex union registration to analyze the social and political organization of homosexuality. It argues that the apparent legal uniformity of Western European evolution hides important and long-term diversities. The argument relies on two types of evidence: same-sex union registration and same-sex union statutes. Please note I will use the o d u io as a general category for same-sex civil partnerships, marriages, Pacs, etc., a d u io egist atio U' as the a t to egiste the u io a iage o partnership registration.

Patterns of same-sex union registration (UR)

Little research has been done on the demographics of same-sex UR in Europe (Anderson et al., 2006;Digoix et al., 2004;[START_REF] Banens | Mariage et partenariat de même sexe en Europe. Vingt ans d e p ie e[END_REF]2011). Some characteristics have been observed in all countries. The first year(s) boom is one of them. The first one or two years after introduction of the statute, enrollment numbers tend to be higher than the following years. This is generally explained by the stock of unregistered couples that had been waiting for the new statute to be introduced, but part of it may also be explained by the intense publicity that generally comes with the legal introduction. For the same reasons, registration numbers may go up again when new major legal changes are introduced, such as when a country that already recognizes same-sex domestic partnerships or civil unions opens the full status of marriage to same-sex couples. And, logically, they may be lower the last year(s) before such changes, because couples may prefer waiting for the new statute that offers increased rights and benefits as compared to options already available to them. Another characteristic that has been observed are the generally lower female UR rates. This reflects differences in the social and psychological organization of male and female homosexuality. Homosexually active men tend to live more frequently in metropolitan areas and show more individualistic and community-oriented behavior than women. Homosexually active women identify more frequently as bisexual and seem more couple-oriented than men. Moreover, male same-sex couples have been estimated as more numerous than female couples. A third observation concerns the diversity of statutes. Surprisingly, different statutes do no appear to be related to registration rates. Civil partnerships may show low UR rates in one country and high in others. The same is true for marriage. Moreover, although the addition of full marriage rights to already existing civil or domestic partnership statutes within a country may boost the UR rates for one or two years, the rates ultimately fall back to where they were previously in all cases.

Regarding UR frequencies, we can distinguish three groups of countries with high within group homogeneity and strong differences between groups. Scandinavia makes up the first group: Denmark, Sweden, Norway, Finland and Iceland. These countries invented the samesex civil partnership in 1989-2002. These were also the first countries to abandon the civil partnership registration at the introduction of same-sex marriage (2009)(2010)(2011)(2012)(2013)(2014). Forerunners in the legal recognition of same-sex couples, as they were in many gender and family issues, the Scandinavian countries do not, however, show high rates of same-sex URs, especially in the case of male couples. The number of annual all-male URs is around 4 for 100 000 male inhabitants. Excluding the first year boom, this number is rather stable over time. When the civil partnership changed to marriage, rates numbers increased by one or two points, but fell back to the initial level soon after. The annual number of all-female URs, on the contrary, has been steadily rising. In all Scandinavian countries, all-female URs started very low (1 to 3 for 100 000 female inhabitants), but have steadily increased since then. Female rates surpassed male rates at the beginning of the century. Their actual level holds between 7 and 9. Female rates are now about twice as high as male rates. No other European country has experienced such differences between male and female rates.

The second group of countries could be called Atlantic: the United Kingdom, the Netherlands, Belgium, France, Spain and Ireland. These countries introduced same-sex UR in the period 1998-20010. They show rather high all-male UR rates, ranging from 9 to 16 for 100 000 male inhabitants (first year booms omitted). Female rates are close to male rates in the UK, the Netherlands and Belgium. They are considerably lower in France, Spain and Ireland, though they are increasing in a Scandinavian-like pattern. Indeed, the Atlantic group is less homogeneous than the Scandinavian group. All countries have in common high male UR rates, at least twice as high as in Scandinavia. But female rates differ. Around the North Sea (UK, Netherlands, Belgium), female rates were almost immediately as high as male rates and have remained so since then. In France, Spain and Ireland, female rates started far below male rates and although they continue to rise, they still trail behind male rates.

France, however, is an outlier. French male UR rates follow a pattern similar to that of women entering URs in Scandinavia. They started relatively low (with lowest value 7.3 in 2001) but have increased ever since, until they reached 16.6 in 2010, which was the highest rate until 2013 when they spiked to 20.6 after the passage of a French bill recognizing samesex marriage. So, from 2008 onwards, France turns out to have higher same-sex UR rates than any other European country, in spite of severe internal criticism of their civil union status known as the Pacs. The reasons for this outlying pattern are unknown.

Central Europe offers the third group of countries: Germany, Austria, the Czech Republic, Hungary and Slovenia. They introduced same-sex civil partnerships in 2001-2010. Very low UR rates are their common characteristic. Male rates are between 0 and 5, female rates are between 0 and 4 for 100 000 inhabitants. Another similarity is that these countries, with the exception of Austria, publish data on UR rates irregularly, usually with long delays, and sometimes not at all, That is the case for Germany, where data have been estimated from annual surveys. In these countries, invisibility seems to be the rule not only for same-sex couples but also for State statistics more broadly. Two countries do not fit these regional classifications. Switzerland has UR rates that are closer to Atlantic than Central European values. Portugal, in contrast, has UR rates that would classify it as a Central European country. Despite these two countries, the demographic analysis offered here cuts along geographical lines: three European regions stand out with distinct UR behavior.

Why different patterns?

Why do different regions show different patterns of same-sex union registration? What motivates cohabiting same-sex couples to register or not? Do same-sex couples cohabit as much in one country as in another? Are there as many same-sex couples in every country? Since [START_REF] Bech | When Men Meet[END_REF], there has been research on all these questions (see for example the works of [START_REF] Festy | La legalization des couples homosexuels en Europe[END_REF][START_REF] Badgett | Sex miscoding and same-sex couple estimates. Evidence from the Fre h Ce sus a d Fa ily Sur ey[END_REF], Banens 2013[START_REF] Rydstrom | Odd Couples. A History of Gay Marriage in Scandinavia, Amsterdam University Press Stambolis-'uhsto fe , Mi hael, La els of lo e: Ho ig a ts egotiate o ot the ultu e of se ual ide tit[END_REF]), but we still lack answers. This paper highlights one aspect of the question: the type of citizenshipin the sense of national belonginggranted to same-sex partners.

The civil partnerships created by Scandinavian countries were restricted exclusively to same-sex couples. They were based on legal and symbolic separateness: marriage remained reserved for different sex couples while civil partnership was the only option for same-sex couples. In some countries (Germany, The Czech Republic, Hungary, Slovenia), this separateness was associated with unequal rights for same-sex couples in the areas of adoption, fiscal and social policy. In other countries (Scandinavia, England and Wales, Switzerland), legal rights and responsibilities came close to marriage, even if some differences remained like the presumption of parenthood and wedding ceremonies. Most importantly, the separate statutes maintained a symbolic difference: gay rights for gay citizens, straight rights for straight citizens. This plural conception of citizenship fitted into certain national political ideologies but not all. That is why the Scandinavian civil partnership was adopted by Germany, Switzerland, the UK and all Central European countries, but not by the Netherlands, Belgium, France, Spain and Portugal.

In 1998, the Netherlands created the first universal statute ge egist ee d pa t e s hap , open to both same-sex and opposite-sex families, though this new legal regime did not offer the full set of marriage rights. The Belgian cohabitation contract was the second, the French Pacs the third. In the following decade, the Netherlands, Belgium and Spain created samesex marriage. In these countries, marriage eventually became a u i e sal statute open to all couples. Though the Dutch partnership, the Belgian cohabitation contract, the French Pacs and the different same-sex marriages do not provide total parenting equality, they do intend to avoid subdividing society into different groups and specific citizenships.

It is important to note that the universalist wave did not start in France but in the Netherlands and that Belgium, Spain, and some years later Portugal, followed the Dutch Figure 1. Blue: universalist statutes, early decriminalization of adult homosexuality, high Union Registration rates. Red: differentialist statutes, late decriminalization, median to low UR rates example, not the French Pacs. Moreover, the differentialist wave did not start in the UK but in Scandinavia and it reached Germany and Switzerland before the UK. European trends in legal recognition of same-sex unions can be described as universalism opposing differentialism, but not as French republican ideology opposing British community-oriented ideology.

As far as homosexuality is concerned, the opposition between universalism and separatism has a long history. The universalist countries decriminalized consenting adult homosexuality at the very beginning of the 19 th century. The differentialist countries decriminalized homosexuality more than one century later, not before 1960 in most countries. Despite common opinion, universalist countries never recriminalized homosexuality. Some countries (the Netherlands in 1911, France in 1942)did raise the age of consent for homosexual actsbut not heterosexual actsto 21 or 18. But higher age of consent for homosexual acts has been the case in both universalist and differentialst countries and has always followed decriminalization. When England and Wales decriminalized adult homosexuality in 1967, the law introduced different ages of consent and this discrimination was not abolished until 2000 (and after judgment of the European Court of Human Rights). Germany decriminalized adult homosexuality in 1969, and introduced different ages of consent that were only abolished twenty years later, in 1989. The main step, of course, was the decriminalization of adult homosexuality and that step occurred in Europe through two distinct waves separated by one and a half centuries.

Quite remarkably, the current recognition of same-sex unions divides Europe exactly along the same dividing line as did decriminalization of adult homosexuality. This shows persisting cultural diversity. The legal reason for this dividing line seems obvious: universalist countries adopted the Napoleonic Code, differentialist countries did not. The Napoleonic Code declared sexuality a private affair, just as religion. Adult consenting sexual practices were decriminalized, with the notable exception of extramarital sexuality of married women. This prohibition was not motivated by sexual or moral considerations, because unmarried women s se ual a ti ities e e ot i i alized. It intended to guarantee that the husband would be the biological father of the children born inside marriage. Sexuality per se was decriminalized, including sodomy in its double signification of homosexuality and zoophilia. At the time, the legislator did not think of homosexuals as a sexual minority, but of homosexuality as a sexual practice. Universalist decriminalization was based on the principle of freedom of sexuality exactly like the separation of state and church was based on the principle of freedom of religion and not on secularization. Both spheres gained freedom in private life, ignored by the State.

Differentialist countries did not decriminalize homosexuality on the principle of freedom but on the principle of equality. The very influential Wolfenden Report (1957) that prepared British decriminalization (1967), stated i e u i e salist te s that ho ose ual behavior between consenting adults in private was part of the "realm of private morality and immorality which is, in brief and crude terms, not the law's business." However, the arguments were inspired by a concept of homosexuality as distinct from heterosexuality; in this concept, homosexual behavior was unable to be changed through criminalization or other means. At the same time, it was not believed to be contagious either. In other words, homosexuality and heterosexuality were assumed to be totally separated. This separateness made it possible to think in terms of gay rights and to concede them. In countries where separate citizenship is politically conceivable, decriminalization of homosexuality and other gay rights have been obtained on the principle of equality.

There was State ignorance at one hand, specific recognition at the other. It happened twice: once for decriminalizing adult homosexuality, once more for recognizing same-sex unions. This shows strong continuity in the political management of sexuality.

The contrast between universalist and separatist Europe has been noticed and analyzed by many scholars at different levels: in ideology [START_REF] Mccaffrey | The Gay Republic, Sexuality, Citizenship and Subversion in France[END_REF][START_REF] Fassin | L i e sio de la uestio ho ose uelle[END_REF], in grass roots politics [START_REF] Fillieule | The global emergence of gay and lesbian politics : national imprints of a worldwide movement[END_REF], and even in personal intimacy [START_REF] Provencher | Queer French: Globalization, Language, and Sexual Citizenship in France[END_REF], Stambolis-Ruhstorfer 2013). It remains an important and meaningful dividing line inside of Europe. But we should bring three amendments to this general idea.

1/ Universalism and differentialism may be more or less dominant in different countries; they are never hegemonic. Many internal debates show this. Outrage is a universalist UK

LGBT movement in a country where separatism is dominant. Irène Théry is an influential differentialist sociologist in France where universalism is dominant. Sweden and Hungary prepared universalist statutes but turned them into differentialist partnerships, etc. However, each country has its special mixture that seems persistent over time.

2/ Universalism is not limited to France. France played a leading role in universalist decriminalization of homosexuality, but it did not in the legal recognition of same-sex unions.

3/ Unlike universalism, there seems to be not one but two differentialisms. One is community-based, identity-based and liberal, producing high visibility. It is bottom-up differentialism, dominant in the UK and, to a lesser extent, in Switzerland. The other is Stateorganized, vertical top-down differentialism, producing less visibility. It is dominant in the Scandinavian and Germanic countries.

Actual UR behavior seems to reflect these long term dividing lines. Top-down differentialist countries may have high social tolerance for homosexuality but gay and lesbian visibility remains rather low and so do UR rates. Bottom-up differentialist countries, on the contrary, show high visibility and higher UR rates. Finally, universalist countries do not show particularly high social visibility, but they do show the highest UR rates. This observation seems well established, even though more research is needed to explain why.

Maks Banens is the author of De Homo-aversie (HUIG, Groningen, 1981) and Vivre avec le VIH (Calmann-Lévy, Paris, 2006) (see References for recent english publications on same-sex unions and same-sex couples)


Július Czap
email: julius.czap@tuke.sk

Stanislav Jendroľ
email: stanislav.jendrol@upjs.sk

František Kardoš
email: frantisek.kardos@upjs.sk

Roman Soták
email: roman.sotak@upjs.sk

Facial parity edge colouring of plane pseudographs

Keywords: plane graph, facial walk, edge colouring 2010 MSC: 05C10, 05C15

A facial parity edge colouring of a connected bridgeless plane graph is such an edge colouring in which no two face-adjacent edges receive the same colour and, in addition, for each face f and each colour c, either no edge or an odd number of edges incident with f is coloured with c. Let χ ′ p (G) denote the minimum number of colours used in such a colouring of G. In this paper we prove that χ ′ p (G) ≤ 20 for any 2-edge-connected plane graph G. In the case when G is a 3-edge-connected plane graph the upper bound for this parameter is 12. For G being 4-edge-connected plane graph we have χ ′ p (G) ≤ 9. On the other hand we prove that some bridgeless plane graphs require at least 10 colours for such a colouring.

Introduction

The famous Four Colour Problem has served as a motivation for many equivalent colouring problems, see e.g. the book of Saaty and Kainen [START_REF] Saaty | The Four-Color Problem, assaults and conquest[END_REF]. The Four Colour Problem was solved in 1976 by Appel and Haken [START_REF] Appel | Every planar map is four colorable[END_REF] (see also Robertson et al. [14] for another proof) and the result is presently known as the Four Colour Theorem (4CT). From the 4CT the following result follows, see [START_REF] Saaty | The Four-Color Problem, assaults and conquest[END_REF]. Theorem 1. The edges of a plane triangulation can be coloured with 3 colours so that the edges bounding every face are coloured distinctly.

In 1965, Vizing [START_REF] Vizing | On an estimate of the chromatic class of a p-graph[END_REF] proved that simple planar graphs with maximum degree at least eight have the edge chromatic number equal to their maximum degree. He conjectured the same if the maximum degree is either seven or six. The first part of this conjecture was proved by Sanders and Zhao in 2001, see [START_REF] Sanders | Planar graphs of maximum degree seven are class I[END_REF]. Note that (also by Vizing) every graph with maximum degree ∆ has the edge chromatic number equal to ∆ or ∆ + 1. These results of Sanders and Zhao and of Vizing can be reformulated in a sense of Theorem 1 in the following way:

Theorem 2. Let G be a 3-edge-connected plane graph with maximum face size ∆ * ≥ 7.

Then the edges of G can be coloured with ∆ * colours in such a way that the edges bounding every face of G are coloured distinctly.

On the other hand, in 1997 Pyber [START_REF] Pyber | Covering the edges of a graph by[END_REF] has shown that the edges of any simple graph can be coloured with at most 4 colours so that all the edges from the same colour class induce a graph with all vertices having odd degree. Mátrai [START_REF] Mátrai | Covering the edges of a graph by three odd subgraphs[END_REF] constructed an infinite sequence of finite simple graphs which require 4 colours in any such colouring. Pyber's result can be stated as follows:

Theorem 3. Let G be a 3-edge-connected plane graph. Then the edges of G can be coloured with at most 4 colours so that for any colour c and any face f of G either no edge or an odd number of edges on the boundary of f is coloured with colour c.

Recently Bunde, Milans, West, and Wu [START_REF] Bunde | Optimal strong parity edge-colouring of complete graphs[END_REF][START_REF] Bunde | Parity and strong parity edge-colouring of graphs[END_REF] introduced a strong parity edge colouring of graphs. It is an edge colouring of a graph G such that each open walk in G uses at least one colour an odd number of times. Let p(G) be the minimum number of colours in a strong parity edge colouring of a graph G. The exact value of p(K n ) for complete graphs is determined in [START_REF] Bunde | Optimal strong parity edge-colouring of complete graphs[END_REF]. They also mention that computing p(G) is NP-hard even when G is a tree.

We say that an edge colouring of a plane graph G is facially proper if no two face-adjacent edges of G receive the same colour. (Two edges are face-adjacent if they are consecutive edges of a facial walk of some face f of G.) Note that colourings in Theorems 1 and 2 are facially proper, but the colouring in Pyber's Theorem 3 need not be facially proper.

Motivated by the parity edge colouring concept introduced by Bunde et al. [START_REF] Bunde | Parity and strong parity edge-colouring of graphs[END_REF] and the above mentioned theorems we define a facial parity edge colouring of a plane graph G as a facially proper edge colouring with the following property: for each colour c and each face f of G either no edge or an odd number of edges incident with f is coloured with the colour c. The problem is to determine for a given bridgeless plane graph G the minimum possible number of colours,

χ ′ p (G), in such a colouring of G. The number χ ′ p (G) is called the facial parity chromatic index of G.
Note that the facial parity chromatic index depends on the embedding of the graph. For example, the graph depicted in Figure 1 has different facial parity chromatic index depending on its embedding. With the embedding on the left, its facial parity chromatic index is 5; whereas with the embedding on the right, its parity chromatic index is 4.

The vertex version of this problem (parity vertex colouring) was introduced in [START_REF] Czap | Parity vertex colouring of plane graphs[END_REF]. The authors proved that every 2-connected plane graph G admits a proper vertex coloring with at most 118 colours such that for each face f and each colour c, either no vertex or an odd number of vertices incident with f is coloured with c. The constant 118 was recently improved to 97 by Kaiser et al. [START_REF] Kaiser | Strong parity vertex coloring of plane graphs[END_REF]. Czap [START_REF] Czap | Parity vertex colouring of outerplane graphs[END_REF] proved that every 2-connected outerplane graph has a parity vertex colouring with at most 12 colours.

In this paper we prove that each connected bridgeless plane graph has a facial parity edge colouring using at most 20 colours, which improves the bound 92 published in [START_REF] Czap | Facial parity edge colouring[END_REF]. The facial parity chromatic index is at most 12 for any 3-edge-connected plane graph. In the case when a plane graph is 4-edge-connected the upper bound is at most 9 for this parameter. We also present graphs which require 10 colours for such a colouring.

Throughout the paper, we mostly use the terminology from a recent book [START_REF] Bondy | Graph theory[END_REF] of Bondy and Murty. All graphs considered are allowed to contain loops and multiedges, unless stated otherwise.

Results

2-edge-connected plane graphs

Let ϕ be a facial parity edge colouring of a bridgeless plane graph G. Observe that in the dual graph G * , the edges of G in each colour class correspond to a factor of G * with the degrees of all the vertices odd or zero, i.e. it is an odd subgraph. Moreover, since ϕ is a facially proper edge colouring in G, it induces a facially proper edge colouring in G * as well.

We say that an edge colouring of a plane graph is odd, if each colour class induces an odd subgraph.

Observation 1. Let G be a plane graph. Then χ ′ p (G) ≤ k if and only if the dual graph G * has a facially proper odd edge colouring using at most k colours. This observation will play a major role in proofs below. Instead of facial parity edge colouring of G we shall investigate facially proper odd edge colouring of G * .

Let us recall the result of Pyber in its original form.

Theorem 4 (Pyber [START_REF] Pyber | Covering the edges of a graph by[END_REF]). The edge set of any simple graph H can be covered by at most 4 edge-disjoint odd subgraphs. Moreover, if H has an even number of vertices then it can be covered by at most 3 edge-disjoint odd subgraphs.

We use this result to establish a general upper bound on facial parity chromatic index for the class of bridgeless plane graphs.

For a face f of a (connected) plane graph G let E(f ) denote the set of edges incident with f . Let ϕ be an edge colouring of a graph G and let c be a colour. Then ϕ -1 (c) = {e ∈ E(G) : ϕ(e) = c} denotes the set of edges coloured with c. Lemma 1. Let G be a connected plane graph. Then there is a facially proper edge colouring ϕ of G using at most 5 colours such that for every two faces f 1 and f 2 of G and every colour

c |ϕ -1 (c) ∩ E(f 1 ) ∩ E(f 2 )| = 0 or |ϕ -1 (c) ∩ E(f 1 ) ∩ E(f 2 )| ≡ 1 (mod 2).
Proof. Let G be a counterexample with minimum number of edges. It is easy to see that G must be 2-connected.

Let M (G) be the medial graph of G: vertices of M (G) correspond to the edges of G; two vertices of M (G) are adjacent if the corresponding edges of G are face-adjacent. Clearly, M (G) is a plane graph, hence, it has a proper vertex colouring ψ M using at most 4 colours.

If any two faces of G share at most one edge, then the colouring ψ of the edges of G, given by the colouring ψ M of the vertices of M (G), has the required property.

Assume that at least two faces, say f 1 and f 2 , share at least two edges. Let e 1 , . . . , e k be the common edges of f 1 and f 2 ordered according to their appearance on the facial walk of f 1 (and f 2 ). Let G 1 , . . . , G k be the components of G \ {e 1 , . . . , e k } such that G i is incident with e i and not incident with e i+1 in G (e k+1 = e 1 ).

If all the graphs G i are singletons (i = 1, 2, . . . , k), then G is a cycle on k vertices and a required colouring can be found easily: Let k = 4ℓ + z, where ℓ is a non-negative integer and z ∈ {2, 3, 4, 5}. We repeat ℓ times the pattern 1, 2, 1, 2 and then use colours 1, 2 . . . , z. The colours 1 and 2 are thus used 2ℓ + 1 times, the remaining (at most three) colours are used once.

Assume that G i has more than one vertex for at least one i ∈ {1, . . . , k}. Suppose k ≥ 3. Let H 0 be a cycle of length k. Let H i be a graph obtained from G i by pasting a path of length 2 to the endvertices of e i-1 and e i (e 0 = e k ). The graphs H 0 , H 1 , . . . , H k have less edges than G, hence, each of them has a required edge 5-colouring, say ϕ i . The colouring ϕ 0 of H 0 can be extended to a colouring ϕ of G in the following way: For each i ∈ {1, . . . , k}, find a permutation of colours used in the colouring ϕ i of H i such that the colours on the edges e i-1 and e i in H 0 and on the corresponding edges in H i coincide; use this colouring for the graph G i . It is easy to see that the colouring ϕ of G obtained this way has the desired property.

Therefore, we may assume that whenever two faces of G share k ≥ 2 edges, then k must be equal to 2. If for some faces f 1 and f 2 and their common edges e 1 and e 2 both the components G 1 and G 2 of G \ {e 1 , e 2 } contain at least 2 vertices, we proceed in the same way as in the previous paragraph. Hence, for each two faces f 1 and f 2 that share two edges, the edges they share are adjacent. But then those two edges receive different colours in the colouring ψ, which implies ψ has the desired property.

Theorem 5. Let G be a 2-edge-connected plane graph. Then

χ ′ p (G) ≤ 20.
Proof. Let ϕ be a facially proper edge colouring of G given by Lemma 1. This colouring induces an edge-decomposition of the dual graph G * into five graphs, say G * i , i = 1, 2, 3, 4, 5. Observe that each edge has an odd multiplicity in every graph G * i , i = 1, 2, 3, 4, 5.

Let H * i be a graph obtained from G * i by simplifying the multiple edges (if two vertices are joined with more than one edge, then we remove all of them but one). The graph H * i is simple, hence, it can be edge-decomposed into at most 4 odd subgraphs (see Theorem 4). Colour each such subgraph of H * i with a distinct colour. To extend this colouring of H * i to a colouring of G * i , colour the multiedges with the same colour as has the corresponding edge in H * i . This way we obtain an edge-decomposition of each G * i into four odd subgraphs; altogether at most 20 colours are used in G * . This colouring of the dual graph G * induces a required colouring of the original graph G, see Observation 1.

Note that there is a graph G such that χ ′ p (G) = 10 and there is a

2-connected graph G ′ such that χ ′ p (G ′ ) = 9.
It is sufficient to consider the graphs depicted in Figure 2.

3-edge-connected plane graphs

In the remaining parts of the paper we investigate facial parity edge colourings of 3-edgeconnected plane graphs. Observe that if G is a 3-edge-connected plane graph, then its dual G * is a simple plane graph. Therefore, we may apply structural properties of planar graphs on the dual graph; in particular, we will use the following one: Theorem 6 (Gonçalves [START_REF] Gonçalves | Edge partition of planar graphs into two outerplanar graphs[END_REF]). Let H = (V, E) be a simple planar graph. Then it has a bipartition of its edge set E = E 1 ∪ E 2 such that the graphs induced by these subsets, H[E 1 ] and H[E 2 ], are outerplanar.

Recall that a (planar) graph is outerplanar if it can be embedded in the plane in such a way that all the vertices are on the boundary of the outer face. Note that for a given plane embedding of a planar graph H, the two outerplanar graphs given by Theorem 6 need not be outerplanarly embedded.

In order to find bounds on the facial parity chromatic index of a 3-edge-connected plane graph, we first decompose its dual into two outerplanar graphs, and then bound the number of colors needed for a facially proper odd edge coloring of them. The structure of outerplanar graphs is given in the following theorem.

Theorem 7 (Borodin, Woodall [START_REF] Borodin | Thirteen colouring numbers for outerplane graphs[END_REF]). If H is a simple outerplanar graph, then at least one of the following cases holds.

There exists an edge uv such that deg

H (u) = deg H (v) = 2.
2. There exists a 3-face uvx such that deg H (u) = 2 and deg H (v) = 3.

There exist two 3-faces xu

1 v 1 and xu 2 v 2 such that deg H (u 1 ) = deg H (u 2 ) = 2, deg H (x) =
4, and these five vertices are all distinct. 4. δ(H) = 1, where δ(H) denotes the minimum vertex degree of H. Lemma 2. Let H be an arbitrary plane embedding of a simple outerplanar graph. Then it has a facially proper odd edge colouring using at most 6 colours.

Proof. Let H be a counterexample with minimum number of edges. Clearly, H is connected. First, we prove several structural properties of H.

Claim 1. H is not a tree.

Proof. If H is a tree, we can find a facially proper odd edge colouring using at most 5 colours as follows: Pick any vertex of H to be the root. We colour the edges of H starting from the root to the leaves. In each step it is sufficient to find a facially proper odd edge colouring of a star with (at most) one precoloured edge. Let S n be a star on n edges e 1 , . . . , e n in a cyclic order. Let {1, 2, 3, 4, 5} be a set of colours. We can assume that the edge e 1 has already been coloured with colour 1. Let n = 4ℓ + z, where ℓ is a non-negative integer and z ∈ {2, 3, 4, 5}. We repeat ℓ times the pattern 1, 2, 1, 2 and then use colours 1, 2, . . . , z. The colours 1 and 2 are thus used 2ℓ + 1 times, the remaining (at most three) colours are used once.

We say that a subgraph H 0 of a connected graph H is hanging on an edge uv if uv is a bridge in H and H 0 is a component of H \ uv.

Claim 2. No tree of order at least two is hanging on any edge of H.

Proof. Let T be a tree hanging on the edge uv, let v ∈ T . Let H ′ = H \ (T \ {v}) be a graph obtained from H by deleting the edges and vertices of T , except for the vertex v.

Clearly, H ′ is outerplanar with less edges than H. Hence, it has a facially proper odd edge colouring using at most 6 colours. We use the same argument as in the proof of the previous claim to extend the colouring of H ′ to a required colouring of H.

Claim 3. There is no edge uv in H such that deg H (u) = 2 and 2 ≤ deg H (v) ≤ 4.
Proof. Let H ′ = H \ uv be a graph obtained from H by deleting the edge uv. The graph H ′ has a required colouring. We can easily extend this colouring to the colouring of H -it suffices to use any colour which does not appear on the edges incident with u or v.

Claim 4. The minimum degree of H is one.

Proof. It follows from Claim 3 and Theorem 7.

Let H ′ be a graph obtained from H by removing all the vertices of degree one. Clearly, H ′ is outerplanar and by Claim 2 the minimum vertex degree of H ′ is at least two. From Theorem 7 it follows that H ′ contains an edge uv such that deg

H ′ (u) = 2 and 2 ≤ deg H ′ (v) ≤ 4.
There is no such edge in H (see Claim 3), hence, in H there are some vertices of degree one adjacent to u or v.

First assume that u is adjacent with some vertices of degree one in H. If it is adjacent with at most three vertices of degree one, then the graph obtained by removing all these vertices is not a counterexample, hence, it has a required colouring. To extend it to a colouring of H, we colour the new edges with colours which do not appear on the edges incident with u.

If u is adjacent with at least four vertices of degree one, then we can find three vertices incident with u, say u 1 , u 2 , x, such that deg H (u 1 ) = deg H (u 2 ) = 1 and the edges uu 1 , ux and uu 2 , ux are face-adjacent. Let us call this configuration a fork. By induction, the graph H \ {uu 1 , uu 2 } has a facially proper odd edge colouring using at most 6 colours. If all the edges incident with u in H \{uu 1 , uu 2 } are coloured with at most four colours then we colour the edges uu 1 , uu 2 with two new colours, else we use the colour which appears on an edge incident with u not face-adjacent to uu 1 nor uu 2 . Now we assume that deg H (u) = 2 and the vertex v is incident with some vertices of degree one in H.

If deg H ′ (v) ≤ 3, then we can use similar arguments as above. Assume that deg H ′ (v) = 4. If v is incident with one or two vertices of degree one in H, it suffices to delete these vertices, apply induction, and use the colour(s) which does not appear on the edges incident with v. If v is incident with at least five vertices of degree one in H, then we can always use the reduction using forks described above.

Suppose that v is incident with precisely three vertices of degree one in H. Let v 1 and v 2 be neighbours of v of degree one such that the edges vv 1 and vv 2 are not face-adjacent. Then, by induction, we find a facially proper odd edge colouring of H \ {v 1 , v 2 } using at most 6 colours. The degree of v in H \ {v 1 , v 2 } is five, therefore, on the five edges incident with v five different colours appear. To extend this colouring to a required colouring of H, we use the colour of the edge incident with v and not face-adjacent to vv 1 nor vv 2 .

Finally, we can suppose that v is incident with precisely four vertices of degree one in H, and that there is no fork incident with v. Let v 1 , . . . , v 8 be the neighbours of v in the cyclic order. Since there is no fork, we may assume that v 1 , v 2 , v 5 , and v 6 are vertices of degree one. Then, by induction, we find a facially proper odd edge colouring of H \ {v 1 , v 2 , v 5 , v 6 } using at most 6 colours. The degree of v in H \ {v 1 , v 2 , v 5 , v 6 } is four, therefore, on the four edges incident with v four different colours appear. To extend this colouring to a required colouring of H, we use the colour of the edge vv 3 to colour vv 1 and vv 6 , and we use the colour of the edge vv 7 to colour vv 2 and vv 5 .

Combining Theorem 6 and Lemma 2 we obtain Theorem 8. Let G be a 3-edge-connected plane graph. Then

χ ′ p (G) ≤ 12. Proof. Let G * be the dual of G. 3-edge-connectedness of G implies that G * is a simple (plane) graph. Let E 1 ∪ E 2 = E(G *
) be the bipartition of the edge set of G * such that the graphs induced by these subsets are outerplanar (see Theorem 6).

Since the graph G * [E i ] induced by the edges from E i is outerplanar, i = 1, 2, it has a facially proper odd edge coloring with at most six colours (see Lemma 2). In this way we obtain a facially proper odd edge colouring of G * with at most 12 colours, therefore the claim follows from Observation 1.

However, this bound does not seem to be best possible.

Conjecture 1. If H is an arbitrary plane embedding of a simple outerplanar graph then it has a facially proper odd edge colouring using at most 5 colours.

It is easy to see that if this conjecture is true, then χ ′ p (G) ≤ 10 for every 3-edge-connected plane graph G.

4-edge-connected plane graphs

For the class of 4-edge-connected plane graphs we use a different approach. The arboricity of a graph is the minimum number of forests into which its edges can be decomposed.

Theorem 9 (Nash-Williams [START_REF] St | Decomposition of finite graphs into forests[END_REF]). Let G be a simple graph. Then the arboricity of G equals max

H⊆G,|V (H)|≥2 |E(H)| |V (H)| -1 ,
where the maximum is taken over all connected subgraphs H on at least two vertices. From this fact and from Theorem 9 it follows that the edges of G we can decomposed into two forests.

Lemma 3. Let H be a simple plane graph. If its arboricity is 2, then there is a decomposition of its edge set into two forests A and B such that each vertex of H is incident with an edge from the forest B.

Proof. Let A 0 and B 0 be two forests such that they form a decomposition of H and the number of vertices which are not incident with any edge of B 0 is the smallest possible. Assume there is a vertex v not incident with any edge of B 0 (it is incident only with A 0 ). Let e be an edge of A 0 incident with v. Let A 1 = A 0 \ {e}, B 1 = B 0 ∪ {e}. Clearly, A 1 and B 1 are forests, A 1 ∪ B 1 = E(H), and the number of vertices not covered by B 1 is smaller than for B 0 , a contradiction.

Theorem 10. Let G be a 4-edge-connected plane graph. Then

χ ′ p (G) ≤ 9.
Proof. Let G * be the dual of G. The graph G is 4-edge-connected, hence, the girth of G * is at least 4.

Let A, B be a decomposition of G * into two forests given by Lemma 3. First we find a facially proper odd edge colouring of B which uses at most 5 colours (see the proof of Claim 1).

For each component C of A we find a facially proper odd edge colouring using at most 4 colours similarly: We root C at any vertex. We proceed from the root to the leaves. It suffices to find facially proper odd edge colouring of stars with (at most) one precoloured edge. Let S n be a star on n edges e 1 , . . . , e n in a cyclic order. Let {1, 2, 3, 4} be a set of colours. We can assume that the edge e 1 has already been coloured with colour 1. Let n = 4ℓ + z, where ℓ is a non-negative integer and z ∈ {2, 3, 4, 5}.

If z = 5, then we use the same colouring as in the proof of Claim 1.

If n = 4ℓ + 5, ℓ ≥ 1, we repeat the pattern 1, 2, 3 three times and then repeat ℓ -1 times the pattern 1, 2, 1, 2.

Let n = 5. The central vertex of the star is incident with at least one edge from B, hence, there are two edges e i , e i+1 that are not face-adjacent in G * . Colour the edges e 1 , . . . , e 5 with three different colours such that the edges e i , e i+1 , e i+3 (indices modulo 5) receive the same colour.

These colourings of A and B together induce a facially proper odd edge colouring of G * using together at most 9 colours, hence, the claim follows from Observation 1.

2

 2

Figure 1 :

 1 Figure 1: Two embeddings of the same graph with different facial parity chromatic index.

Figure 2 :

 2 Figure 2: Examples of graphs with no facial parity edge colouring using less than 10 and 9 colours, respectively.

Corollary 1 . 4 .

 14 Let G = (V, E, F ) be a simple plane graph with girth at least 4. Then its arboricity is at most two. Proof. Observe that any connected subgraph H (on at least 3 vertices) of G also has girth at least From the Euler's polyhedral formula |V (H)|-|E(H)|+|F (H)| = 2 and from the fact 2 • |E(H)| = f ∈F (H) deg H (f ) ≥ 4 • |F (H)| we can easily derive that |E(H)| ≤ 2 • |V (H)| -4.

 [START_REF] Appel | Every planar map is four colorable[END_REF]This work was supported by the Slovak Science and Technology Assistance Agency under the contract No. APVV-0023-10, by the Slovak VEGA Grant 1/0428/10, and by the Agency of the Slovak Ministry of Education for the Structural Funds of the EU, under project ITMS:26220120007.