Approximating the Longest Cycle Problem on Graphs with Bounded Degree

• Published in: Computing and Combinatorics pp. 870 - 884
• Main Authors: Chen, Guantao, Gao, Zhicheng, Yu, Xingxing, Zang, Wenan
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2005; Springer
• Series: Lecture Notes in Computer Science
• Subjects: Applied sciences; Computer science; control theory; systems; Exact sciences and technology; Flows in networks. Combinatorial problems; Information retrieval. Graph; Operational research and scientific management; Operational research. Management science; Theoretical computing; Connected graph; Vertex(graph); Combinatorial problem; Cycle(graph)
• ISBN: 9783540280613; 3540280618
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/11533719_88

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}.