Abstract
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)?A such that for every directed edge uv from u to v, c(u)?c(v) ? 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 ? 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)?V(H 2)=? and V(H 1)?V(H 2)=V(G). Then ? g (G)≤min{max{? g (H 1), max v ? V(H 2) deg(v,G)+1},max{? g (H 2), max u ? V(H 1) deg (u, G) + 1}}. We also show that this bound is best possible. (2) If G is a simple graph without a K 3,3-minor, then ? g (G)≤5.
Published Version
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