 
Summary: Long cycles in critical graphs
Noga Alon
Michael Krivelevich
Paul Seymour
Abstract
It is shown that any kcritical graph with n vertices contains a cycle of length at least
2 log(n  1)/ log(k  2), improving a previous estimate of Kelly and Kelly obtained in 1954.
1 Introduction
A graph is kcritical if its chromatic number is k but the chromatic number of any proper subgraph
of it is at most k  1. For a graph G, let L(G) denote the maximum length of a cycle in G, and
define Lk(n) = min L(G) where the minimum is taken over all kcritical graphs G with at least n
vertices. Answering a problem of Dirac, Kelly and Kelly [3] proved that for every fixed k > 2 the
function Lk(n) tends to infinity as n tends to infinity. They also showed that L4(n) O(log2
n), and
after several improvements by Dirac and Read, Gallai [2] proved that for every fixed k 4 there are
infinitely many values of n for which
Lk(n)
2(k  1)
log(k  2)
log n.
