# Coloring squares of graphs with mad constraints

## TL;DR

This paper investigates coloring the squares of graphs with constraints on maximum average degree and maximum degree, proving new bounds on choosability and correspondence coloring, and providing counterexamples to previous conjectures.

## Contribution

It establishes that graphs with mad less than 4 and large maximum degree have their squares be (3Δ+1)-choosable and correspondence-colorable, and constructs graphs with high chromatic number.

## Key findings

- Squares of such graphs are (3Δ+1)-choosable.
- Constructed graphs with mad<4 and high Δ have chromatic number at least 2.5Δ.
- Improves previous bounds on chromatic number for these graph classes.

## Abstract

A proper vertex $k$-coloring of a graph $G=(V,E)$ is an assignment $c:V\to \{1,2,\ldots,k\}$ of colors to the vertices of the graph such that no two adjacent vertices are associated with the same color. The square $G^2$ of a graph $G$ is the graph defined by $V(G)=V(G^2)$ and $uv \in E(G^2)$ if and only if the distance between $u$ and $v$ is at most two. We denote by $\chi(G^2)$ the chromatic number of $G^2$, which is the least integer $k$ such that a $k$-coloring of $G^2$ exists. By definition, at least $\Delta(G)+1$ colors are needed for this goal, where $\Delta(G)$ denotes the maximum degree of the graph $G$. In this paper, we prove that the square of every graph $G$ with $\text{mad}(G)<4$ and $\Delta(G) \geqslant 8$ is $(3\Delta(G)+1)$-choosable and even correspondence-colorable. Furthermore, we show a family of $2$-degenerate graphs $G$ with $\text{mad}(G)<4$, arbitrarily large maximum degree, and $\chi(G^2)\geqslant \frac{5\Delta(G)}{2}$, improving the result of Kim and Park.