On the maximum number of edges in -critical graphs

A graph is called $k$-critical if its chromatic number is $k$ but every 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 every integer $k\geq 4$. Using a structural characterisation of Greenwell and Lovász 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 in 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.