List Set Colouring: Bounds and Algorithms

Abstract

For an integer $b \geq 1$, the $b$-choice number of a graph $G$ is the minimum integer $k$ such that, for every assignment of a set $S(v)$ of at least $k$ colours to each vertex $v$ of $G$, there is a $b$-set colouring of $G$ that assigns to each vertex $v$ a $b$-set $B(v) \subseteq S(v) \; (|B(v)|=b)$ so that adjacent vertices receive disjoint $b$-sets. This is a generalization of the notions of choice number and chromatic number of a graph. Using probabilistic arguments, we show that, for some positive constant $c > 0$ (independent of $b$), the $b$-choice number of any graph $G$ on $n$ vertices is at most $c (b\chi) (\ln (n/\chi)+1)$ where $\chi = \chi(G)$ denotes the chromatic number of $G$. For any fixed $b$, this bound is tight up to a constant factor for each $n,\chi$. This generalizes and extends a result of Noga Alon [1]wherein a similar bound was obtained for 1-choice numbers of complete $\chi$-partite graphs with each part having size $n/\chi$. We also show that the proof arguments are constructive, leading to polynomial time algorithms for the list colouring problem on certain classes of graphs, provided each vertex is given a list of sufficiently large size.