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

Distance-two coloring of sparse graphs

Consider a graph $G = (V, E)$ and, for each vertex $v \in V$, a subset $\Sigma(v)$ of neighbors of $v$. A $\Sigma$-coloring is a coloring of the elements of $V$ so that vertices appearing together in some $\Sigma(v)$ receive pairwise distinct colors. An obvious lower bound for the minimum number of colors in such a coloring is the maximum size of a set $\Sigma(v)$, denoted by $\rho(\Sigma)$. In this paper we study graph classes $F$ for which there is a function $f$, such that for any graph $G \in F$ and any $\Sigma$, there is a $\Sigma$-coloring using at most $f(\rho(\Sigma))$ colors. It is proved that if such a function exists for a class $F$, then $f$ can be taken to be a linear function. It is also shown that such classes are precisely the classes having bounded star chromatic number. We also investigate the list version and the clique version of this problem, and relate the existence of functions bounding those parameters to the recently introduced concepts of classes of bounded expansion and nowhere-dense classes.Comment: 13 pages - revised versio