关于图的3-条件色数

On 3-Hued Coloring of Graphs

中文关键词:  $r$-条件色数  $3$-正常图  三角.

英文关键词:$r$-hued chromatic number  $3$-normal graph  triangle.

中文摘要:

      对整数$k>0$, $r>0$,图$G$的一个$(k,r)$-染色是$G$的顶点的一个正常$k$-染色,使得$G$中任意度数为$d$的顶点$v$,其邻域中至少出现$\min\{d,r\}$种不同的颜色.使$G$有一个正常的$(k,r)$-染色的最小$k$值称为$G$的条件色数,记为$\chi_r(G)$.如果 $\chi_r(G)=\chi(G)$,~则称图$G$ 是$r$-正常的.本文主要给出了$3$-正常图的两个充分条件和特殊图类的$3$-条件色数的最好上界.

英文摘要:

      For integers $k>0$, $r>0$, a $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices such that every vertex of degree $d$ is adjacent to vertices with at least $\min\{d,r\}$ different colors. The $r$-hued chromatic number, denoted by $\chi_r(G)$, is the smallest integer $k$ for which a graph $G$ has a $(k,r)$-coloring. Define a graph $G$ is $r$-normal, if $\chi_r(G)=\chi(G)$. In this paper, we present two sufficient conditions for a graph to be $3$-normal, and the best upper bound of $3$-hued chromatic number of a certain families of graphs.