On Group Choosability of Graphs, II

Given a group A and a directed graph G, let F(G, A) denote the set of all maps f : E(G) → A. Fix an orientation of G and a list assignment L : V (G) → 2A. For an f ∈ F(G, A), G is (A, L , f )-colorable if there exists a map c : V (G) → ∪v∈V (G)L(v) such that c(v) ∈ L(v), ∀v ∈ V (G) and c(x)− c(y) = f (xy) for every edge e = xy directed from x to y. If for any f ∈ 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) → 2A, then G is k-group choosable. The group choice number, denoted by χgl(G), is the minimum k such that G is kgroup 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 K5 or a K3,3 as a minor, and discuss group choosability versions of Hadwiger’s and Woodall’s conjectures. H. Chuang · G. R. Omidi · N. Zakeri Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111, Isfahan, Iran H.-J. Lai College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, Xinjiang, Peoples Republic of China H.-J. Lai (B) · K. Wang Department of Mathematics, West Virginia University, Morgantown, WV 26505, USA e-mail: [email protected] G. R. Omidi School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5746, Tehran, Iran