The list r-hued coloring of Halin graph

Abstract

Let $L$ be a list assignment of colors available for vertices of a graph $G$. An $(L,r)$-coloring of a graph $G$ is a proper coloring $c$ such that every vertex $v\in V\left(G\right)$ receives at least $min\{d_G\left(v\right),r\}$ colors in its neighbors and $c\left(v\right)\in L\left(v\right)$. The list $r$-hued chromatic number ${\chi }_{L,r}\left(G\right)$ of a graph $G$, is the smallest $k$ such that for each list assignment $L$ satisfying $\left|L\left(v\right)\right|=k$, for any $v\in V\left(G\right)$, $G$ has an $(L,r)$-coloring. An $(L,r)$-coloring is called an $r$-hued $k$-coloring if $L\left(v\right)=\{1,2,\dots ,k\}$ for all $v\in V\left(G\right)$. Let $G$ be a Halin graph. We determine the upper bounds of the list $r$-hued chromatic number of Halin graphs when $r=2$ and $r=\Delta$. This improves former results on 2-hued coloring of Halin graphs.