A note on total and list edge-colouring of graphs of tree-width 3

It is shown that Halin graphs are $\Delta$-edge-choosable and that graphs of tree-width 3 are $(\Delta+1)$-edge-choosable and $(\Delta +2)$-total-colourable.Comment: arXiv admin note: substantial text overlap with arXiv:1504.0212

A Unified Approach to Distance-Two Colouring of Graphs on Surfaces

In this paper we introduce the notion of $\Sigma$-colouring of a graph $G$: For given subsets $\Sigma(v)$ of neighbours of $v$, for every $v\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that appear together in some $\Sigma(v)$ receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of graphs embedded in a surface. We prove a general result for graphs embeddable in a fixed surface, which implies asymptotic versions of Wegner's and Borodin's Conjecture on the planar version of these two colourings. Using a recent approach of Havet et al., we reduce the problem to edge-colouring of multigraphs, and then use Kahn's result that the list chromatic index is close to the fractional chromatic index. Our results are based on a strong structural lemma for graphs embeddable in a fixed surface, which also implies that the size of a clique in the square of a graph of maximum degree $\Delta$ embeddable in some fixed surface is at most $\frac32\,\Delta$ plus a constant.Comment: 36 page

Thresholds and the structure of sparse random graphs

In this thesis, we obtain approximations to the non-3-colourability threshold of sparse random graphs and we investigate the structure of random graphs near the region where the transition from 3-colourability to non-3-colourability seems to occur. It has been observed that, as for many other properties, the property of non-3-colourability of graphs exhibits a sharp threshold behaviour. It is conjectured that there exists a critical average degree such that when the average degree of a random graph is around this value the probability of the random graph being non-3-colourable changes rapidly from near 0 to near 1. The difficulty in calculating the critical value arises because the number of proper 3-colourings of a random graph is not concentrated: there is a `jackpot' effect. In order to reduce this effect, we focus on a sub-family of proper 3-colourings, which are called rigid 3-colourings. We give precise estimates for their expected number and we deduce that when the average degree of a random graph is bigger than 5, then the graph is asymptotically almost surely not 3-colourable. After that, we investigate the non-$k$-colourability of random regular graphs for any $k \geq 3$. Using a first moment argument, for each $k \geq 3$ we provide a bound so that whenever the degree of the random regular graph is bigger than this, then the random regular graph is asymptotically almost surely not $k$-colourable. Moreover, in a (failed!) attempt to show that almost all 5-regular graphs are not 3-colourable, we analyse the expected number of rigid 3-colourings of a random 5-regular graph. Motivated by the fact that the transition from 3-colourability to non-3-colourability occurs inside the subgraph of the random graph that is called the 3-core, we investigate the structure of this subgraph after its appearance. Indeed, we do this for the $k$-core, for any $k \geq 2$; and by extending existing techniques we obtain the asymptotic behaviour of the proportion of vertices of each fixed degree. Finally, we apply these results in order to obtain a more clear view of the structure of the 2-core (or simply the core) of a random graph after the emergence of its giant component. We determine the asymptotic distributions of the numbers of isolated cycles in the core as well as of those cycles that are not isolated there having any fixed length. Then we focus on its giant component, and in particular we give the asymptotic distributions of the numbers of 2-vertex and 2-edge-connected components

Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes

A quadrangulation is a graph embedded on the sphere such that each face is bounded by a walk of length 4, parallel edges allowed. All quadrangulations can be generated by a sequence of graph operations called vertex splitting, starting from the path P_2 of length 2. We define the degree D of a splitting S and consider restricted splittings S_{i,j} with i <= D <= j. It is known that S_{2,3} generate all simple quadrangulations. Here we investigate the cases S_{1,2}, S_{1,3}, S_{1,1}, S_{2,2}, S_{3,3}. First we show that the splittings S_{1,2} are exactly the monotone ones in the sense that the resulting graph contains the original as a subgraph. Then we show that they define a set of nontrivial ancestors beyond P_2 and each quadrangulation has a unique ancestor. Our results have a direct geometric interpretation in the context of mechanical equilibria of convex bodies. The topology of the equilibria corresponds to a 2-coloured quadrangulation with independent set sizes s, u. The numbers s, u identify the primary equilibrium class associated with the body by V\'arkonyi and Domokos. We show that both S_{1,1} and S_{2,2} generate all primary classes from a finite set of ancestors which is closely related to their geometric results. If, beyond s and u, the full topology of the quadrangulation is considered, we arrive at the more refined secondary equilibrium classes. As Domokos, L\'angi and Szab\'o showed recently, one can create the geometric counterparts of unrestricted splittings to generate all secondary classes. Our results show that S_{1,2} can only generate a limited range of secondary classes from the same ancestor. The geometric interpretation of the additional ancestors defined by monotone splittings shows that minimal polyhedra play a key role in this process. We also present computational results on the number of secondary classes and multiquadrangulations.Comment: 21 pages, 11 figures and 3 table

Progress on the adjacent vertex distinguishing edge colouring conjecture

A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree $\Delta$ and no isolated edge has an adjacent vertex distinguishing edge colouring with $\Delta + 300$ colours, provided $\Delta$ is large enough. We show that this bound can be reduced to $\Delta + 19$. This is motivated by the conjecture of Zhang, Liu, and Wang (2002) that $\Delta + 2$ colours are enough for $\Delta \geq 3$.Comment: v2: Revised following referees' comment