若干图的点强全着色

The Vertex Strong Total Coloring of Some Graphs

图G(V,E)的一正常k-全着色σ称为G(V,E)的一个k-点强全着色,当且仅当v∈V(G),N[v]中的元素着不同颜色,其中N[v]={u|vu∈E(G)}∪{v}.并且vχsT(G)=min{k|存在G的一个k-点强全着色}称为G(V,E)的点强全色数.本文得到了一些特殊图的点强全色数χvTs(G),并提出猜想:对于简单图G,有k(G)≤χvTs(G)≤k(G)+1,这里k(G)表示图G中所有顶点间距离不超过2的点集的最大顶点数.

A proper k-total coloring a of graph G（V,E） is called a k-vertex strong total coloring of G（V,E） if and only if for arbitary v∈V（G）, the elements in N[v] are colored with different colors, where N[v] = {u｜ vu∈E（G）} U {v} and χT^vs （G） =min{k｜there is a k-vertex strong total coloring of G} is called the vertex strong total chromatic number of G. In this paper, we obtain the vertex strong total chromatic number χT^vs （G） of some special graphs and present a conjecture; For simple graph G, has k（G）≤χT^vs（G）≤k（G）＋ 1, where k（G） denotes the maximum value of the element of all such vertices set where the distance between each two vertices is at most 2.