Group colorability of multigraphs

Let G be a multigraph with a fixed orientation D and let Gamma be a group. Let F(G, Gamma) denote the set of all functions f : E(G) -> Gamma. A multigraph G is Gamma-colorable if and only if for every f is an element of F(G, Gamma), there exists a Gamma-coloring c : V(G) -> Gamma such that for every e = uv is an element of E(G) (assumed to be directed from u to v), c(u)c(v)(-1) not equal f(e). We define the group chromatic number chi(g)(G) to be the minimum integer m such that G is Gamma-colorable for any group Gamma of order >= m under the orientation D. In this paper, we investigate the properties of chi(g) for multigraphs and prove an analogue to Brooks' Theorem. (C) 2012 Elsevier B.V. All rights reserved.