List recoloring of planar graphs

Abstract

A list assignment $L$ of a graph $G$ is a function that assigns to every vertex $v$ of $G$ a set $L\left(v\right)$ of colors. A proper coloring $\alpha$ of $G$ is called an $L$-coloring of $G$ if $\alpha \left(v\right)\in L\left(v\right)$ for every $v\in V\left(G\right)$. For a list assignment $L$ of $G$, the $L$-recoloring graph $G(G,L)$ of $G$ is a graph whose vertices correspond to the $L$-colorings of $G$ and two vertices of $G(G,L)$ are adjacent if their corresponding $L$-colorings differ at exactly one vertex of $G$. A $d$-face (resp. $d^{-}$-face) in a plane graph is a face of length (resp. at most) $d$. Dvořák and Feghali conjectured for a planar graph $G$ and a list assignment $L$ of $G$: (i) If $\left|L\left(v\right)\right|\ge 10$ for every $v\in V\left(G\right)$, then the diameter of $G(G,L)$ is $O\left(\left|V\left(G\right)\right|\right)$. (ii) If $G$ is triangle-free and $\left|L\left(v\right)\right|\ge 7$ for every $v\in V\left(G\right)$, then the diameter of $G(G,L)$ is $O\left(\left|V\left(G\right)\right|\right)$. In a recent paper, Cranston (2022) has proved (ii). In this paper, we prove the following results.

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  Let $G$ be a plane graph without adjacent 3-faces and $L$ be a list assignment of $G$ such that $\left|L\left(v\right)\right|\ge 9$ for every $v\in V\left(G\right)$, then the diameter of $G(G,L)$ is at most $13\left|V\left(G\right)\right|$.
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  Let $G$ be a plane graph in which 3-faces are not adjacent to $5^{-}$-faces and $L$ be a list assignment of $G$ such that $\left|L\left(v\right)\right|\ge 7$ for every $v\in V\left(G\right)$, then the diameter of $G(G,L)$ is at most $242\left|V\left(G\right)\right|$. This result confirms (ii) on a superclass of the class of triangle-free planar graphs.

As an additional result, we show that if the independence number of a $k$-colorable graph $G$ is at most $p\ge 2$ and $L$ is a list assignment of $G$ such that $L\left(v\right)=\{1,2,\dots ,\ell \}$ for every $v\in V\left(G\right)$, where $\ell \ge \lfloor \frac{p\cdot k}{2}\rfloor +1$, then the diameter of $G(G,\ell )$ is at most $4\left|V\left(G\right)\right|$. We further show that if $\ell <\lfloor \frac{p\cdot k}{2}\rfloor +1$, then $G(G,L)$ can be a disconnected graph.