Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.27111 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917536537247744 |
|---|---|
| author | Miao, Yu Liu, Fengxia |
| author_facet | Miao, Yu Liu, Fengxia |
| contents | 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, $\min\{r,Δ(G)\}+1\leqχ_{L,r}(G)\leq\min\{r,Δ(G)\}+2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27111 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The list r-hued coloring of trees and unicyclic graphs Miao, Yu Liu, Fengxia Combinatorics 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, $\min\{r,Δ(G)\}+1\leqχ_{L,r}(G)\leq\min\{r,Δ(G)\}+2$. |
| title | The list r-hued coloring of trees and unicyclic graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.27111 |