Longest Cycle above Erdős-Gallai Bound
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In 1959, Erdős and Gallai proved that every graph G with average vertex degree ad(G)≥2 contains a cycle of length at least ad(G). We provide an algorithm that for k≥0 in time 2^O(k) n^O(1) decides whether a 2-connected n-vertex graph G contains a cycle of length at least ad(G)+k. This resolves an open problem explicitly mentioned in several papers. The main ingredients of our algorithm are new graph-theoretical results interesting on their own.
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