Group colorability of multigraphs

Let G be a multigraph with a fixed orientation D and let Γ be a group. Let F(G,Γ) denote the set of all functions f:E(G)→Γ. A multigraph G is Γ-colorable if and only if for every f∈F(G,Γ), there exists a Γ-coloring c:V(G)→Γ such that for every e=uv∈E(G) (assumed to be directed from u to v), c(u)c(v)−1≠f(e). We define the group chromatic number χg(G) to be the minimum integer m such that G is Γ-colorable for any group Γ of order ≥m under the orientation D. In this paper, we investigate the properties of χg for multigraphs and prove an analogue to Brooks' Theorem.