On group chromatic number of graphs

Let G be a graph and A an Abelian group. Denote by F(G, A) the set of all functions from E(G) to A. Denote by D an orientation of E(G). For f epsilon F(G,A), an (A,f)-coloring of G under the orientation D is a function c : V(G) bar right arrow A such that for every directed edge uv from u to v, c(u)-c(v) not equal f(uv). G is A-colorable under the orientation D if for any function f is an element of F(G, A), G has an (A, f)-coloring. It is known that A-colorability is independent of the choice of the orientation. The group chromatic number of a graph G is defined to be the least positive integer m for which G is A-colorable for any Abelian group A of order >= m, and is denoted by chi(g)(G). In this note we will prove the following results. (1) Let H-1 and H-2 be two subgraphs of G such that V(H-1) boolean AND V(H-2) = 0 and V(H-1) boolean OR V(H-2)=V(G). Then chi(g)(G) <= min{max{chi(g)(H-1), max(v is an element of V(H2)) deg(v,G) + 1}, max{chi(g)(H-2), max(u is an element of V(H1)) deg (u, G) + 1}}. We also show that this bound is best possible. (2) If G is a simple graph without a K-3,K-3-minor, then chi(g)(G) <= 5.