On the density of critical graphs with no large cliques

TL;DR

This paper establishes new lower bounds on the average degree of large, critical graphs that do not contain large cliques, advancing understanding of graph coloring properties in such structures.

Contribution

It proves a lower bound on the average degree of $K_{ ext{omega}+1}$-free $L$-critical graphs with bounded clique size, improving previous bounds for graphs without large cliques.

Findings

Lower bound on average degree for large critical graphs without large cliques.

Implication that list-chromatic number is bounded by a function of maximum average degree and clique size.

Results hold for sufficiently large $k$ and very small epsilon.

Abstract

A graph $G$ is \textit{$k$-critical} if $\chi(G) = k$ and every proper subgraph of $G$ is $(k - 1)$-colorable, and if $L$ is a list-assignment for $G$, then $G$ is \textit{$L$-critical} if $G$ is not $L$-colorable but every proper induced subgraph of $G$ is. In 2014, Kostochka and Yancey proved a lower bound on the average degree of an $n$-vertex $k$-critical graph tending to $k - \frac{2}{k - 1}$ for large $n$ that is tight for infinitely many values of $n$, and they asked how their bound may be improved for graphs not containing a large clique. Answering this question, we prove that for $\varepsilon \leq 2.6\cdot10^{-10}$, if $k$ is sufficiently large and $G$ is a $K_{\omega + 1}$-free $L$-critical graph where $\omega \leq k - \log^{10}k$ and $L$ is a list-assignment for $G$ such that $|L(v)| = k - 1$ for all $v\in V(G)$, then the average degree of $G$ is at least $(1 + \varepsilon)(k - 1) - \varepsilon \omega - 1$. This result implies that for some $\varepsilon > 0$, for every graph $G$ satisfying $\omega(G) \leq \mathrm{mad}(G) - \log^{10}\mathrm{mad}(G)$ where $\omega(G)$ is the size of the largest clique in $G$ and $\mathrm{mad}(G)$ is the maximum average degree of $G$, the list-chromatic number of $G$ is at most $\left\lceil (1 - \varepsilon)(\mathrm{mad}(G) + 1) + \varepsilon\omega(G)\right\rceil$.