Longest Cycle above Erdős-Gallai Bound

Abstract

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.