完全二部图K8,n(472≤n≤980)的点可区别E-全染色

Vertex-distinguishing E-total coloring of complete bipartite graph K 8,n with 472≤n≤980

图G的一个E-全染色是指使相邻点染以不同颜色且每条关联边与它的端点染以不同颜色的全染色.对图G的一个E-全染色f,一旦u,v∈V(G),u≠v,就有C(u)≠C(v),其中C(x)表示在f下点x的颜色以及与x关联的边的色所构成的集合,则f称为图G的点可区别的E-全染色,简称为VDET染色.令χe v t(G)=min{k|G存在k-VDET染色},称χe v t(G)为图G的点可区别E-全色数.在该文中,利用组合分析法、反证法并构造具体染色,讨论给出了完全二部图K 8,n(472≤n≤980)的点可区别E-全色数.

Let G be a simple graph.Also,a total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color,and no edge of G receives the same color as one of its endpoints.For an E-total coloring f of a graph G,if C(u)≠C(v)for any two distinct vertices u and v of V(G),where C(x)denotes the set of colors of vertex x and of edges incident with x under f,then f is called a vertex-distinguishing E-total coloring of G.Letχe v t(G)=min{k|G has a k-VDET coloring}.Thenχe v t(G)is called the VDET chromatic number of G.The VDET coloring of complete bipartite graph K 8,n is discussed and the VDET chromatic number of K 8,n(472≤n≤980)has been obtained by using combinatorial analysis method,contradiction and constructing concrete coloring.