Color-Critical Graphs Have Logarithmic Circumference
Combinatorics
2011-04-14 v2
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.
Keywords
Cite
@article{arxiv.0908.3169,
title = {Color-Critical Graphs Have Logarithmic Circumference},
author = {Asaf Shapira and Robin Thomas},
journal= {arXiv preprint arXiv:0908.3169},
year = {2011}
}