Color-Critical Graphs Have Logarithmic Circumference

Asaf Shapira; Robin Thomas

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arXiv · 0908.3169

Color-Critical Graphs Have Logarithmic Circumference

Asaf Shapira·Robin Thomas

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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.

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BibTeX RIS

Asaf Shapira, Robin Thomas. 2011-04-13. Color-Critical Graphs Have Logarithmic Circumference. https://arxiv.org/abs/0908.3169

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