On the Density of Critical Graphs with No Large Cliques

Abstract

A graph G is 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 L-critical if G is not L-colorable but every proper 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 there exists some $\varepsilon > 0$ for which the following holds. If k is sufficiently large and G is a $K_{\omega + 1}$-free L-critical graph where $\omega \le k - \log ^{10}k$ 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 $\omega (G) \le \textrm{mad}(G) - \log ^{10}\textrm{mad}(G)$ where $\omega (G)$ is the size of the largest clique in G and $\textrm{mad}(G)$ is the maximum average degree of G, the list-chromatic number of G is at most .