几类操作图的全控制染色

Total Dominator Chromatic Number with Several Types of Operations Graphs

图G的一个全控制染色是图G的一个正常点染色,使得G中每个顶点v控制(除了v以外)的至少一个色类。图G的全控制染色所需的最少颜色数称为G的全控制色数,记为χ_(d)^(t)(G)。本文研究Middle图和Total图的内部构造,分别给出了一般图G的Middle图M^(*)(G)的全控制色数χ_(d)^(t) (M^(*)(G))的上下界;分别计算了完全图K_(n)、完全二部图K_(n,m)和双星图S_(n,m)的Middle图M^(*)(K_(n))、M^(*)(K_(n,m))和M^(*)(S_(n,m))的全控制色数以及得到了完全图K_(n)和双星图S_(n,m)的Total图T(K_(n))和T(S_(n,m))的全控制色数。

A total dominator coloring of a graph G is the proper vertex coloring so that each vertex v of G dominates except v at least one color class.The minimum number of colors required for a total dominator coloring of G is called the total dominator chromatic number of G,denoted byχ_(d)^(t)(G).Meanwhile,in this paper,by analyzing the structural properties of the Middle graph and Total graph,Middle graph M*(K_(n)),M*(K_(n,m))and M*(S_(n,m))of the complete graph K_(n),the complete bipartite graph K_(n,m) and double star graphs S_(n,m) are calculated values of the total dominator chromatic number and obtained exact value of the total dominator chromatic number of the Total graph T(K_(n))and T(S_(n,m))of double star graph S_(n,m) and complete graph K_(n).