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arXiv 2209.05992math.COcs.DM
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可平面图的列表赋环

List recoloring of planar graphs

L. Sunil Chandran, Uttam K. Gupta, Dinabandhu Pradhan

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摘要：

图$G$的列表分配$L$是将颜色集合$L(v)$分配给$G$的每个顶点$v$的函数。如果对每个$v\in V(G)$都有$α(v)\in L(v)$，则称$G$的一个正常着色$α$为$L$ -着色.对于$G$的列表分配$L$，$G$的$L$ -着色图$\mathcal{G}(G,L)$是其顶点对应于$G$的$L$ -着色的图，并且如果$\mathcal{G}(G,L)$的两个顶点对应的$L$ -着色在$G$的恰好一个顶点处不同，则它们相邻。平面图中的$d$ -面是长度为$d$的面。Dvotakák和Feghali证明了平面图$G$和$G$的列表分配$L$：（i）若对每个$v\in V(G)$都有$|L(v)|\geq 10$，则$\mathcal{G}(G,L)$的直径为$O(|V(G)|)$ . (ii)如果$G$是无三角形的，并且每个$v\in V(G)$都有$|L(v)|\geq 7$，则$\mathcal{G}(G,L)$的直径为$O(|V(G)|)$。在最近的一篇论文中，Cranston（European J. Combin.（2022）证明了（2）。在本文中，我们证明了以下结果。设$G$是一个平面图，$L$是$G$的列表赋值. $\bullet$如果对于$G$的每个$3$ -面，最多有两个$3$ -面与之相邻，并且对于每个$v\in V(G)$有$|L(v)|\geq 10$，则$\mathcal{G}(G,L)$的直径最多为$190|V(G)|$。$\bullet$如果对于$G$的每个$3$ -面，最多有一个$3$ -面与之相邻，并且对于每个$v\in V(G)$有$|L(v)|\geq 9$，则$\mathcal{G}(G,L)$的直径最多为$13|V(G)|$。$\bullet$如果与任何$3$ -面相邻的面对于每个$v\in V(G)$具有至少$6$和$|L(v)|\geq 7$的长度，则$\mathcal{G}(G,L)$的直径至多为$242|V(G)|$。这一结果加强了Cranston关于（ii）的结果。

英文摘要：

A list assignment $L$ of a graph $G$ is a function that assigns to every vertex $v$ of $G$ a set $L(v)$ of colors. A proper coloring $α$ of $G$ is called an $L$-coloring of $G$ if $α(v)\in L(v)$ for every $v\in V(G)$. For a list assignment $L$ of $G$, the $L$-recoloring graph $\mathcal{G}(G,L)$ of $G$ is a graph whose vertices correspond to the $L$-colorings of $G$ and two vertices of $\mathcal{G}(G,L)$ are adjacent if their corresponding $L$-colorings differ at exactly one vertex of $G$. A $d$-face in a plane graph is a face of length $d$. Dvořák and Feghali conjectured for a planar graph $G$ and a list assignment $L$ of $G$, that: (i) If $|L(v)|\geq 10$ for every $v\in V(G)$, then the diameter of $\mathcal{G}(G,L)$ is $O(|V(G)|)$. (ii) If $G$ is triangle-free and $|L(v)|\geq 7$ for every $v\in V(G)$, then the diameter of $\mathcal{G}(G,L)$ is $O(|V(G)|)$. In a recent paper, Cranston (European J. Combin. (2022)) has proved (ii). In this paper, we prove the following results. Let $G$ be a plane graph and $L$ be a list assignment of $G$. $\bullet$ If for every $3$-face of $G$, there are at most two $3$-faces adjacent to it and $|L(v)|\geq 10$ for every $v\in V(G)$, then the diameter of $\mathcal{G}(G,L)$ is at most $190|V(G)|$. $\bullet$ If for every $3$-face of $G$, there is at most one $3$-face adjacent to it and $|L(v)|\geq 9$ for every $v\in V(G)$, then the diameter of $\mathcal{G}(G,L)$ is at most $13|V(G)|$. $\bullet$ If the faces adjacent to any $3$-face have length at least $6$ and $|L(v)|\geq 7$ for every $v\in V(G)$, then the diameter of $\mathcal{G}(G,L)$ is at most $242|V(G)|$. This result strengthens the Cranston's result on (ii).

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