折叠立方体图的邻点可区别全色数(英文)

The Adjacent Vertex-distinguishing Total Chromatic Number of Folded Hypercubes

简单图G的全染色是指对G的点和边都进行染色.称全染色为正常的如果没有相邻或关联元素染同一种颜色.简单图G=(V,E)的正常全染色h称为它的邻点可区别全染色如果对任意两个相邻顶点u、v,有H(u)≠H(u),其中H(u)={h(u)}∪{h(uw)|uw∈E(G)}而H(v)={h(v)}∪{h(vx)|vx∈E(G)}.G的邻点可区别全染色所需最少颜色数称为G邻点可区别全色数,记为χat(G).本文考虑折叠立方体图FQ_n的邻点可区别全色数,证明了对任意n≥2,有χat(FQ_n)=n+3.

A total coloring of a simple graph G is a coloring of both edges and vertices. A total coloring is proper if no two adjacent or incident elements receive the same color. An adjacent vertex-distinguishing total coloring h of a simple graph G = （Y,E） is a proper total coloring of G such that H（u）≠ H（v） for any two adjacent vertices u and v, where H（u） = {h（u）} U） （h（uw）｜uw ∈ E（G）} and H（v） = {h（v）} U {h（vx）[vx ∈ E（G）}. The minimum number of colors required for an adjacent vertex-distinguishing total coloring of G is called the adjacent vertex- distinguishing total chromatic number of G and denoted by Xat（G）. In this paper, we consider the adjacent vertex-distinguishing total chromatic number of the folded hypercubes FQn and prove that Xat（FQn） = n ＋ 3 for n ≥ 2.