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Abstract: A graph G is k-critical if every proper subgraph of G is k-1-colorable, butthe graph G itself is not. We prove that every k-critical graph on n verticeshas 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 thebound cannot be improved to exceed 2k-1log n-logk-2. We thus settle theproblem of bounding the minimal circumference of k-critical graphs, raised byDirac in 1952 and Kelly and Kelly in 1954.



Author: Asaf Shapira, Robin Thomas

Source: https://arxiv.org/







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