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 = u v ∈ E ( G ) (assumed to be directed from u to v ), c ( u ) c ( v ) − 1 ≠ f ( e ) . We define the group chromatic number χ g ( 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 χ g for multigraphs and prove an analogue to Brooks’ Theorem.