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Longest Cycle above Erd\H{o}s-Gallai Bound

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arxiv 2202.03061 v1 pith:QCZ4PHMH submitted 2022-02-07 cs.DS cs.DMmath.CO

classification cs.DScs.DMmath.CO
keywords cyclealgorithmcontainsgraphleastlengthaboveaverage
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In 1959, Erd\H{o}s and Gallai proved that every graph G with average vertex degree ad(G)\geq 2 contains a cycle of length at least ad(G). We provide an algorithm that for k\geq 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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