The list r-hued coloring of P5-free graph

Abstract

Given a list $L$ of graph $G$, an $(L,r)$-coloring of graph $G$ is a proper coloring such that the color of vertex $v$ belongs to its list $L\left(v\right)$, and each vertex of degree $d_G\left(v\right)$ is adjcent to vertices with at least $\min \{r,d_G\left(v\right)\}$ different colors. The list $r$-hued chromatic number, 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. We prove the following: $\left(i\right)$ If $G=G(U,V)$ is a connected $P_5$-free bipartite graph and min $\left\{\right|U|,|V\left|\right\}=s$, then ${\chi }_{L,r}\left(G\right)\le \max \{2r,$ $s+1\}$. $\left(ii\right)$ If $G$ is a connected $P_5$-free graph and $r\le 3$, then ${\chi }_{L,r}\left(G\right)\le r{\chi }_L\left(G\right)$.