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 integerm 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. © 2012 Elsevier B.V. All rights reserved.