The Parameterised Complexity of List Problems on Graphs of Bounded Treewidth

We consider the parameterised complexity of several list problems on graphs, with parameter treewidth or pathwidth. In particular, we show that List Edge Chromatic Number and List Total Chromatic Number are fixed parameter tractable, parameterised by treewidth, whereas List Hamilton Path is W[1]-hard, even parameterised by pathwidth. These results resolve two open questions of Fellows, Fomin, Lokshtanov, Rosamond, Saurabh, Szeider and Thomassen (2011).Comment: Author final version, to appear in Information and Computation. Changes from previous version include improved literature references and restructured proof in Section

Goldberg's Conjecture is true for random multigraphs

In the 70s, Goldberg, and independently Seymour, conjectured that for any multigraph $G$, the chromatic index $\chi'(G)$ satisfies $\chi'(G)\leq \max \{\Delta(G)+1, \lceil\rho(G)\rceil\}$, where $\rho(G)=\max \{\frac {e(G[S])}{\lfloor |S|/2\rfloor} \mid S\subseteq V \}$. We show that their conjecture (in a stronger form) is true for random multigraphs. Let $M(n,m)$ be the probability space consisting of all loopless multigraphs with $n$ vertices and $m$ edges, in which $m$ pairs from $[n]$ are chosen independently at random with repetitions. Our result states that, for a given $m:=m(n)$, $M\sim M(n,m)$ typically satisfies $\chi'(G)=\max\{\Delta(G),\lceil\rho(G)\rceil\}$. In particular, we show that if $n$ is even and $m:=m(n)$, then $\chi'(M)=\Delta(M)$ for a typical $M\sim M(n,m)$. Furthermore, for a fixed $\varepsilon>0$, if $n$ is odd, then a typical $M\sim M(n,m)$ has $\chi'(M)=\Delta(M)$ for $m\leq (1-\varepsilon)n^3\log n$, and $\chi'(M)=\lceil\rho(M)\rceil$ for $m\geq (1+\varepsilon)n^3\log n$.Comment: 26 page

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

Terminal-Pairability in Complete Bipartite Graphs

We investigate the terminal-pairibility problem in the case when the base graph is a complete bipartite graph, and the demand graph is also bipartite with the same color classes. We improve the lower bound on maximum value of $\Delta(D)$ which still guarantees that the demand graph $D$ is terminal-pairable in this setting. We also prove a sharp theorem on the maximum number of edges such a demand graph can have.Comment: 8 pages, several typos correcte