Some results on (a:b)-choosability

A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not necessarily $(c:d)$-choosable. Applying probabilistic methods, an upper bound for the $k^{th}$ choice number of a graph is given. We also prove that a directed graph with maximum outdegree $d$ and no odd directed cycle is $(k(d+1):k)$-choosable for every $k \geq 1$. Other results presented in this article are related to the strong choice number of graphs (a generalization of the strong chromatic number). We conclude with complexity analysis of some decision problems related to graph choosability

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

Defective 3-Paintability of Planar Graphs

A $d$-defective $k$-painting game on a graph $G$ is played by two players: Lister and Painter. Initially, each vertex is uncolored and has $k$ tokens. In each round, Lister marks a chosen set $M$ of uncolored vertices and removes one token from each marked vertex. In response, Painter colors vertices in a subset $X$ of $M$ which induce a subgraph $G[X]$ of maximum degree at most $d$. Lister wins the game if at the end of some round there is an uncolored vertex that has no more tokens left. Otherwise, all vertices eventually get colored and Painter wins the game. We say that $G$ is $d$-defective $k$-paintable if Painter has a winning strategy in this game. In this paper we show that every planar graph is 3-defective 3-paintable and give a construction of a planar graph that is not 2-defective 3-paintable.Comment: 21 pages, 11 figure

Linear colorings of subcubic graphs

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list assigned to that vertex and the neighbors of every degree-two vertex receive different colors, unless the graph is $C_5$ or $K_{3,3}$. This confirms a conjecture raised by Esperet, Montassier, and Raspaud. Our proof is constructive and yields a linear-time algorithm to find such a coloring