Group Number Chromatic Of Graphs

Let G=(V,E) be a graph and A be a non-trivial Abelian group, and let F(G,A) denote the set of all functions f:E(G)→A.Denote by D an orientation of E(G).Then G is A-colorable if and only if for every f(?)F(G,A) there exists an A-coloring c:V(G)→A such that for every e=xy(?E)(G) (assumed to be directed from x to y), c(x)-c(y)(?)f(e).If G is a graph, we define its group chromatic number Xg (G) to be the minimum number m for which G is A-colorable for any Abelian group A of order s m under the orientation D. This paper is mainly concerned with group chromatic number and some results are given.Give a grah G let k= maxHδ(H) where the maximum is taken over all spanned subgraphs H of G, then xg (G)≤k+1.The cartesian product of simple graphs G and H is the graph G□H whose vertex is V(G)×V(H) and whose edge set is the set if all pairs (u1,v1)(u2,v2) such that either (u1,u2)EE(G) and v1=v2, or v1 v2 EE(H) and u1,=u2. Thus, we can clearly gets,Xg (Pn□Pm)=3,Xg,(Pn□Cm)≤4,Xg(Cn□Cm)≤4,Xg(Qn)≤n-1. This paper principally proves that, (1) let A≥3 and let G be a graph such that△(G)≤△and K△+1(?) G, then xg (G)≤△; (2)Let G1 and G2 be simple graphs, then Xg(G1□G2)≤max{Xg(G,)+1,Xg(G2)+1}.