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Long cycles in critical graphs Michael Krivelevich

Summary: Long cycles in critical graphs
Noga Alon
Michael Krivelevich
Paul Seymour
It is shown that any k-critical 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 k-critical 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 k-critical 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
2(k - 1)
log(k - 2)
log n.


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics