Approximating the longest cycle problem on graphs with bounded degree

Abstract

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 Ω(nlogd-12), 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 Ω(nlogb2) in G, where b = max{64, 4d + 1}. Our result substantially improves the best existing bound Ω(n log2(d-1)2+12). © Springer-Verlag Berlin Heidelberg 2005.