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.