A 2-Factor in a Graph and the Number of Its Components

• Project/Area Number: 19500017
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Fundamental theory of informatics
• Research Institution: Nihon University
• Principal Investigator: SAITO Akira Nihon University, 文理学部, 教授 (90186924)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2007: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: グラフ / ハミルトンサイクル / 因子 / 成分 / 禁止部分グラフ / 最小次数 / 2-因子 / 連結 / 連結度 / Chvatal-Erdos条件 / 独立数 / 次数条件

Research Abstract

In this research, we interpreted a hamiltonian cycle as a 2-factor with one component, and investigated conditions which differentiate the existence of a hamiltonian cycle and that of a 2-factor, or more generally, which give information on the number of components in a 2-factor. We chose Dirac's condition, the Chvatal-Erdos condition and conditions based on forbidden subgraphs. We discovered that while neither Dirac's condition nor the Chvatal-Erdos condition gives any information on the number of components in a 2-factor, there is a notable difference between forbidden subgraphs forcing the existence of a hamiltonian cycle and those forcing the existence of a 2-factor.

Research Products

• [Journal Article] Closure, stability and iterated line graphs with a 2-factor (2009)
  • Author(s): L. Xiong, A. Saito
  • Journal Title: Discrete Mathematics Vol.309 Pages: 5000-5010
  • Peer Reviewed
• [Journal Article] Two forbidden subgraphs and the existence of a 2-factor in graphs (2009)
  • Author(s): R.E.L. Aldred, J. Fujisawa, A. Saito
  • Journal Title: Australasian Journal of Combinatorics Vol.44 Pages: 235-246
  • Peer Reviewed
• [Journal Article] Two forbidden subgraphs and the existence of a 2-factor in graphs (2009)
  • Author(s): R.E.L Aldred
  • Journal Title: Australasian Journal of Combinatorics 44 Pages: 235-246
  • Peer Reviewed
• [Journal Article] Closure, stability and iterated line graphs with a 2-factor (2009)
  • Author(s): L.Xiong
  • Journal Title: Discrete Mathematics 309 Pages: 5000-5010
  • Peer Reviewed
• [Journal Article] Chvatal-Erdos Theorem-old theorem with new aspects- (2008)
  • Author(s): A. Saito
  • Journal Title: Lecture Notes in Computer Science Vol.4535 Pages: 191-200
  • Peer Reviewed
• [Journal Article] independence and 2-domination in bipartite graphs (2008)
  • Author(s): J. Fujisawa, A. Hansberg, T. Kubo, A. Saito, M. Sugita, L. Volkmann
  • Journal Title: Australasian Journal of Combinatorics Vol.40 Pages: 265-268
  • Peer Reviewed
• [Journal Article] Independence and 2-domination in bipartite graphs (2008)
  • Author(s): J. Fujisawa
  • Journal Title: Australasian Journal of Combinatorics 40 Pages: 265-268
  • Peer Reviewed
• [Journal Article] Chvatal-Erdos Theorem---0ld theorem with new aspects--- (2008)
  • Author(s): A. Saito
  • Journal Title: Lecture Notes in Computer Science 4535 Pages: 191-200
  • Peer Reviewed
• [Journal Article] The Chvatal-Erdos condition and 2-factors with a specified number of components (2007)
  • Author(s): G. Chen, R.J. Gould, K. Kawarabayashi, K. Ota, A. Saito, I. Schiermeyer
  • Journal Title: Discussiones Mathematicae Graph Theory Vol.27 Pages: 401-407
  • Peer Reviewed
• [Journal Article] Edge-dominating cycles in graphs (2007)
  • Author(s): S. Fujita, A. Saito, T. Yamashita
  • Journal Title: Discrete Mathematics Vol.307 Pages: 2934-3942
  • Peer Reviewed
• [Journal Article] Reduction for 3-connected graphs of minimum degree at least four (2007)
  • Author(s): S. Bau, A. Saito
  • Journal Title: Graphs and Combinatorics Vol.23 Pages: 135-144
  • NAID: 40007363258
  • Peer Reviewed
• [Journal Article] Graphs with small boundary (2007)
  • Author(s): Y. Hasegawa, A. Saito
  • Journal Title: Discrete Mathematics Vol.307 Pages: 1801-1807
  • Peer Reviewed
• [Journal Article] Degree conditions for hamiltonicity: counting the number of missing edges (2007)
  • Author(s): R.J. Faudree, A. Saito, R.H. Schelp, I. Schiermeyer
  • Journal Title: Discrete Mathematics Vol.307 Pages: 873-877
  • Peer Reviewed
• [Journal Article] Graphs with small boundary (2007)
  • Author(s): Y. Hasegawa
  • Journal Title: Discrete Mathematics 307 Pages: 1801-1807
  • Peer Reviewed
• [Journal Article] Reduction for 3-connected graphs of minimum degree at least four (2007)
  • Author(s): S. Bau
  • Journal Title: Graphs and Combinatorics 23(Supplement) Pages: 135-144
  • Peer Reviewed
• [Journal Article] Edge-dominating cycles in graphs (2007)
  • Author(s): S. Fujita
  • Journal Title: Discrete Mathematics 307 Pages: 2934-2942
  • Peer Reviewed
• [Presentation] Cycles of length modulo k(招待講演) (2009)
  • Author(s): 斎藤明
  • Organizer: Paths, Cycles and Graph Structures Workshop
  • Place of Presentation: University of Colorado, Denver, Colorado, U.S.A.
  • Year and Date: 2009-07-12
• [Presentation] Cycles of length modulo k (2009)
  • Author(s): Akira Saito
  • Organizer: Paths, Cycles and Graph Structures Workshop
  • Place of Presentation: University of Colorado, Denver, U.S.A.
  • Year and Date: 2009-07-12
• [Presentation] Forbidden subgraphs and 2-factors(招待講演) (2009)
  • Author(s): 斎藤明
  • Organizer: SIAM Annual Meeting
  • Place of Presentation: Colorado Convention Center, Colorado, Denver, U.S.A.
  • Year and Date: 2009-07-10
• [Presentation] Forbidden subgraphs and 2-factors (2009)
  • Author(s): Akira Saito
  • Organizer: 2009 SIAM Annual Meeting
  • Place of Presentation: Colorado Convention Center, Denver, U.S.A.
  • Year and Date: 2009-07-10
• [Presentation] Bipartite Graphs(招待講演) (2008)
  • Author(s): 斎藤明
  • Organizer: The second International Conference on Mathematics and Natural Sciences
  • Place of Presentation: Bandung Institute of Technology, Bandung, Indonesia
  • Year and Date: 2008-10-28
• [Presentation] Bipartite graphs (招待講演) (2008)
  • Author(s): 斎藤 明
  • Organizer: The 2^<nd> International Conference on Mathematics and Natural Sciences 2008
  • Place of Presentation: Bandung Institute ofTechnology, Bandung, Indonesia
  • Year and Date: 2008-10-28
• [Presentation] Forbidden subgraphs and the existence of a 2-factor (2008)
  • Author(s): 斎藤明
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-24
• [Presentation] Forbidden subgraphs and matchings in graphs(招待講演) (2008)
  • Author(s): 斎藤明
  • Organizer: 21st. Cumberland Conference on Graph Theory Combinatorics and Computing
  • Place of Presentation: Vanderbilt University, Nashville, Tennessee, U.S.A.
  • Year and Date: 2008-05-15
• [Presentation] Forbidden subgraphs and matchings in graphs (招待講演) (2008)
  • Author(s): 斎藤 明
  • Organizer: 21st. Cumberland Conference on GraphTheory, Combinatorics and Computing
  • Place of Presentation: Vanderbilt University, Nashville, U.S.A.
  • Year and Date: 2008-05-15
• [Presentation] Forbidden subgraphs and matchings in edge-deleted graphs (2008)
  • Author(s): 斎藤明
  • Organizer: 日本数学会年会
  • Place of Presentation: 近畿大学
  • Year and Date: 2008-03-23
• [Presentation] Closure and small cutsets (2007)
  • Author(s): 斎藤明
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-21
• [Presentation] Chvatal-Erdos theorem - Old theorem with new aspects - (2007)
  • Author(s): 斎藤 明
  • Organizer: Kyoto International Conference on Computational Geometry and Graph Theory(招待講演)
  • Place of Presentation: 京都大学
  • Year and Date: 2007-06-14
• [Presentation] Chvatal-Erdos Theorem(招待講演) (2007)
  • Author(s): 斎藤明
  • Organizer: Kyoto International Conference on Computational Geometry and Graph Theory
  • Place of Presentation: 京都大学
  • Year and Date: 2007-05-15