On the maximum number of edges in k-critical graphs

A graph is called $k$-critical if its chromatic number is $k$ but any proper subgraph has chromatic number less than $k$. An old and important problem in graph theory asks to determine the maximum number of edges in an $n$-vertex $k$-critical graph. This is widely open for any integer $k\geq 4$. Using a structural characterization of Greenwell and Lov\'asz and an extremal result of Simonovits, Stiebitz proved in 1987 that for $k\geq 4$ and sufficiently large $n$, this maximum number is less than the number of edges in the $n$-vertex balanced complete $(k-2)$-partite graph. In this paper we obtain the first improvement on the above result in the past 35 years. Our proofs combine arguments from extremal graph theory as well as some structural analysis. A key lemma we use indicates a partial structure in dense $k$-critical graphs, which may be of independent interest.