On the Density of Critical Graphs with No Large Cliques

graph G is k -critical if χ (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 - 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 ε > 0 for which the following holds. If k is sufficiently large and G is a K_ω + 1 -free L -critical graph where ω≤ k - log ^10k and L is a list assignment for G such that |L(v)| = k - 1 for all v∈ V(G) , then the average degree of G is at least (1 + ε )(k - 1) - εω - 1 . This result implies that for some ε > 0 , for every graph G satisfying ω (G) ≤mad(G) - log ^10mad(G) where ω (G) is the size of the largest clique in G and mad(G) is the maximum average degree of G , the list-chromatic number of G is at most .