On Group Choosability of Graphs, II

Abstract

Given a group A and a directed graph G, let F(G, A) denote the set of all maps ${f : E(G) \rightarrow A}$. Fix an orientation of G and a list assignment ${L : V(G) \mapsto 2^A}$. For an ${f \in F(G, A)}$, G is (A, L, f)-colorable if there exists a map ${c:V(G) \mapsto \cup_{v \in V(G)}L(v)}$ such that ${c(v) \in L(v)}$, ${\forall v \in V(G)}$ and ${c(x)-c(y)\neq f(xy)}$ for every edge e = xy directed from x to y. If for any ${f\in F(G,A)}$, G has an (A, L, f)-coloring, then G is (A, L)-colorable. If G is (A, L)-colorable for any group A of order at least k and for any k-list assignment ${L:V(G) \rightarrow 2^A}$, then G is k-group choosable. The group choice number, denoted by ${\chi_{gl}(G)}$, is the minimum k such that G is k-group choosable. In this paper, we prove that every planar graph is 5-group choosable, and every planar graph with girth at least 5 is 3-group choosable. We also consider extensions of these results to graphs that do not have a K ₅ or a K _3,3 as a minor, and discuss group choosability versions of Hadwiger’s and Woodall’s conjectures.