The list r-hued coloring of trees and unicyclic graphs

Abstract

Let $r$ be a positive integer and $G$ be a graph. The list $r$-hued chromatic number of $G$, denoted by $χ_{L,r}(G)$, is the smallest integer $k$, such that for each $k$-list $L$ of $G$, $G$ has an $(L,r)$-coloring. It is proved in [Discrete Math. 306 (16) (2006) 1997-2004] that every tree $G$ satisfies $χ_{r}(G)=\min\{r,Δ(G)\}+1$. It is known that every cycle graph $C_{n}$ with order $n$ has $χ_{L,r}(C_{n})=χ_{r}(C_{n})$. The main results are the following: $(1)$ If $G$ is a tree, then $χ_{L,r}(G)=\min\{r,Δ(G)\}+1$; $(2)$ Let $G$ be a unicyclic graph which is not isomorphic to the cycle $C_{n}$. If $n\neq 5$ and $r\geq3$, then $χ_{L,r}(G)=\min\{r,Δ(G)\}+1$; otherwise, $\min\{r,Δ(G)\}+1\leqχ_{L,r}(G)\leq\min\{r,Δ(G)\}+2$.