On the maximum number of edges in k-critical graphs

TL;DR

This paper improves the upper bound on the maximum number of edges in large $k$-critical graphs, combining extremal graph theory and structural analysis to advance understanding of graph coloring properties.

Contribution

It provides the first improvement in 35 years on the known upper bound for edges in large $k$-critical graphs, using new structural insights.

Findings

Established a new upper bound for edges in large $k$-critical graphs.

Identified a key structural lemma for dense $k$-critical graphs.

Combined extremal and structural methods for graph analysis.

Abstract

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.