Upper bound of the list r-hued chromatic number

Abstract

Let $k$, $r$ be positive integers. For a color list $L$ on $V\left(G\right)$, if $\left|L\left(v\right)\right|=k$ for any $v\in V\left(G\right)$, then $L$ is a $k$-list of $G$. Given a list $L$ of $G$, an $(L,r)$-coloring of $G$ is a proper vertex coloring $c$ such that $c\left(v\right)\in L\left(v\right)$ and any vertex is adjacent to vertices with at least min$\{r,d_G\left(v\right)\}$ different colors. The list $r$-hued chromatic number of $G$, denoted by ${\chi }_{L,r}\left(G\right)$, is the smallest integer $k$ such that for any $k$-list $L$ of $G$, $G$ has an $(L,r)$-coloring. In this paper, we prove that if $G$ is a connected graph, then ${\chi }_{L,r}\left(G\right)\le min\{r\Delta \left(G\right),{\Delta }^2\left(G\right)\}+1$, where the equality holds if and only if $r\ge \Delta \left(G\right)$ and $G$ is a Moore graph with diameter 2.