On Group Chromatic Number of Graphs |
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Authors: | Hong-Jian Lai Xiangwen Li |
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Institution: | (1) Department of Mathematics, West Virginia University, Morgantown, WV 26505, USA;(2) Department of Mathematics, Huazhong Normal University, Wuhan, 430079, China |
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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 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. |
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