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 ∈ F (G,A), an (A, f)-coloring of G under the orientation D is a function c : V (G) 7→ A such that for every directed edge uv from u to v, c(u) − c(v) 6= f(uv). G is A-colorable under the orientation D if for any function f ∈ 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 χg(G). In this note we will prove the following results. (1) Let H1 and H2 be two subgraphs of G such that V (H1) ∩ V (H2) = ∅ and V (H1) ∪ V (H2) = V (G). Then χg(G) ≤ min{max{χg(H1),maxv∈V (H2) deg(v,G)+1},max{χg(H2),maxu∈V (H1) deg(u,G)+1}}. We also show that this bound is best possible. (2) If G is a simple graph without a K3,3-minor, then χg(G) ≤ 5.