Coloring the squares of graphs whose maximum average degrees are less than 4

TL;DR

This paper investigates the chromatic number of the square of graphs with maximum average degree less than 4, providing new bounds, counterexamples to existing conjectures, and extending previous results in graph coloring theory.

Contribution

It improves bounds on the list chromatic number of graph squares under certain conditions and disproves Charpentier's conjecture through explicit counterexamples.

Findings

For $c extgreater 1$, if $mad(G) extless 4 - 1/c$ and $ riangle(G) extgreater 14c-7$, then $ ext{ch}_ ext{ell}(G^2) extless 2 riangle(G)$.

Counterexamples show graphs with $mad(G) extless 4$ and $ riangle(G) extgreater D$ where $ ext{chi}(G^2) extgreater 2 riangle(G) + 2$.

Disproves Charpentier's conjecture on the chromatic number of graph squares for certain parameters.

Abstract

The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. The {\em maximum average degree} of $G$, $mad (G)$, is the maximum among the average degrees of the subgraphs of $G$. It is known in \cite{BLP-14-JGT} that there is no constant $C$ such that every graph $G$ with $mad(G)< 4$ has $\chi(G^2) \leq \Delta(G) + C$. Charpentier \cite{Charpentier14} conjectured that there exists an integer $D$ such that every graph $G$ with $\Delta(G)\ge D$ and $mad(G)<4$ has $\chi(G^2) \leq 2 \Delta(G)$. Recent result in \cite{BLP-DM} implies that $\chi(G^2) \leq 2 \Delta(G)$ if $mad(G) < 4 -\frac{1}{c}$ with $\Delta(G) \geq 40c -16$. In this paper, we show for $c\ge 2$, if $mad(G) < 4 - \frac{1}{c}$ and $\Delta(G) \geq 14c-7$, then $\chi_\ell(G^2) \leq 2 \Delta(G)$, which improves the result in \cite{BLP-DM}. We also show that for every integer $D$, there is a graph $G$ with $\Delta(G)\ge D$ such that $mad(G)<4$, and $\chi(G^2) \geq 2\Delta(G) +2$, which disproves Charpentier's conjecture. In addition, we give counterexamples to Charpentier's another conjecture in \cite{Charpentier14}, stating that for every integer $k\ge 3$, there is an integer $D_k$ such that every graph $G$ with $mad(G)<2k$ and $\Delta(G)\ge D_k$ has $\chi(G^2) \leq k\Delta(G) -k$.