Color-Critical Graphs Have Logarithmic Circumference

TL;DR

This paper proves that k-critical graphs on n vertices have a cycle length at least proportional to log n, resolving a longstanding problem about their minimal circumference and improving previous bounds.

Contribution

It establishes a new lower bound on the circumference of k-critical graphs, settling a problem posed in the mid-20th century.

Findings

Every k-critical graph on n vertices has a cycle of length at least log n/(100 log k)

The bound cannot be improved beyond 2(k-1) log n / log(k-2)

The result settles a problem posed by Dirac and Kelly in the 1950s

Abstract

A graph G is k-critical if every proper subgraph of G is (k-1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least log n/(100log k), improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that the bound cannot be improved to exceed 2(k-1)log n/log(k-2). We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954.