An improved procedure for colouring graphs of bounded local density

We develop an improved bound for the chromatic number of graphs of maximum degree $\Delta$ under the assumption that the number of edges spanning any neighbourhood is at most $(1-\sigma)\binom{\Delta}{2}$ for some fixed $0<\sigma<1$. The leading term in this bound is best possible as $\sigma\to0$. As two consequences, we advance the state of the art in two longstanding and well-studied graph colouring conjectures, the Erd\H{o}s-Ne\v{s}et\v{r}il conjecture and Reed's conjecture. We prove that the strong chromatic index is at most $1.772\Delta^2$ for any graph $G$ with sufficiently large maximum degree $\Delta$. We prove that the chromatic number is at most $\lceil 0.801(\Delta+1)+0.199\omega\rceil$ for any graph $G$ with clique number $\omega$ and sufficiently large maximum degree $\Delta$.Comment: 21 page

Strong chromatic index of sparse graphs

A coloring of the edges of a graph $G$ is strong if each color class is an induced matching of $G$. The strong chromatic index of $G$, denoted by $\chi_{s}^{\prime}(G)$, is the least number of colors in a strong edge coloring of $G$. In this note we prove that $\chi_{s}^{\prime}(G)\leq (4k-1)\Delta (G)-k(2k+1)+1$ for every $k$-degenerate graph $G$. This confirms the strong version of conjecture stated recently by Chang and Narayanan [3]. Our approach allows also to improve the upper bound from [3] for chordless graphs. We get that $$ for any chordless graph $G$. Both bounds remain valid for the list version of the strong edge coloring of these graphs

Strong edge-colouring of sparse planar graphs

A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching. It is known that every planar graph with maximum degree $\Delta$ has a strong edge-colouring with at most $4\Delta+4$ colours. We show that $3\Delta+1$ colours suffice if the graph has girth 6, and $4\Delta$ colours suffice if $\Delta\geq 7$ or the girth is at least 5. In the last part of the paper, we raise some questions related to a long-standing conjecture of Vizing on proper edge-colouring of planar graphs

A note on coloring vertex-transitive graphs

We prove bounds on the chromatic number $\chi$ of a vertex-transitive graph in terms of its clique number $\omega$ and maximum degree $\Delta$. We conjecture that every vertex-transitive graph satisfies $\chi \le \max \left\{\omega, \left\lceil\frac{5\Delta + 3}{6}\right\rceil\right\}$ and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with $\Delta \ge 13$ we prove the Borodin-Kostochka conjecture, i.e., $\chi\le\max\{\omega,\Delta-1\}$

Distance colouring without one cycle length

We consider distance colourings in graphs of maximum degree at most $d$ and how excluding one fixed cycle length $\ell$ affects the number of colours required as $d\to\infty$. For vertex-colouring and $t\ge 1$, if any two distinct vertices connected by a path of at most $t$ edges are required to be coloured differently, then a reduction by a logarithmic (in $d$) factor against the trivial bound $O(d^t)$ can be obtained by excluding an odd cycle length $\ell \ge 3t$ if $t$ is odd or by excluding an even cycle length $\ell \ge 2t+2$. For edge-colouring and $t\ge 2$, if any two distinct edges connected by a path of fewer than $t$ edges are required to be coloured differently, then excluding an even cycle length $\ell \ge 2t$ is sufficient for a logarithmic factor reduction. For $t\ge 2$, neither of the above statements are possible for other parity combinations of $\ell$ and $t$. These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).Comment: 14 pages, 1 figur

New bounds for the max-kk-cut and chromatic number of a graph

We consider several semidefinite programming relaxations for the max-$k$-cut problem, with increasing complexity. The optimal solution of the weakest presented semidefinite programming relaxation has a closed form expression that includes the largest Laplacian eigenvalue of the graph under consideration. This is the first known eigenvalue bound for the max-$k$-cut when $k>2$ that is applicable to any graph. This bound is exploited to derive a new eigenvalue bound on the chromatic number of a graph. For regular graphs, the new bound on the chromatic number is the same as the well-known Hoffman bound; however, the two bounds are incomparable in general. We prove that the eigenvalue bound for the max-$k$-cut is tight for several classes of graphs. We investigate the presented bounds for specific classes of graphs, such as walk-regular graphs, strongly regular graphs, and graphs from the Hamming association scheme

Defective and Clustered Graph Colouring

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric