  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.
