Approximating the Longest Cycle Problem on Graphs with Bounded Degree

In 1993, Jackson and Wormald conjectured that if G is a 3-connected n-vertex graph with maximum degree d≥ 4 then G contains a cycle of length $\Omega(n^{\log_{d-1}2})$, and showed that this bound is best possible if true. In this paper we present an O(n3) algorithm for finding a cycle of length $\Omega(n^{\log_{b}2})$ in G, where b= max {64,4d+1}. Our result substantially improves the best existing bound $\Omega(n^{\log_{2(d-1)^2+1}2})$.