Color-critical graphs have logarithmic circumference

Asaf Shapira, Robin Thomas

Research output: Contribution to journalArticlepeer-review

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 logn/(100logk), 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)logn/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.

Original languageEnglish
Pages (from-to)2309-2326
Number of pages18
JournalAdvances in Mathematics
Volume227
Issue number6
DOIs
StatePublished - 20 Aug 2011
Externally publishedYes

Keywords

  • Connectivity
  • Critical graphs
  • Long cycles

All Science Journal Classification (ASJC) codes

  • General Mathematics

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