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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Omega(n^{\log_{d-1}2})$ \end{document}, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Omega(n^{\log_{b}2})$ \end{document} in G, where b = max {64,4d + 1}. Our result substantially improves the best existing bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Omega(n^{\log_{2(d-1)^2+1}2})$ \end{document}.