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2002 Fiscal Year Final Research Report Summary

Research on Factors of Graphs Using Closure Operations

Research Project

Project/Area Number 13640138
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionNIHON UNIVERSITY

Principal Investigator

SAITO Akira  Nihon Univ., Dept. Computer Science and System Analysis, Professor, 文理学部, 教授 (90186924)

Project Period (FY) 2001 – 2002
Keywordsclosure / factor / degree-sum / Ryjacek closure / factor-critical / extendable
Research Abstract

The purpose of this research was to apply closure operations, which had been used in the study of hamiltonian cycles, to factor theory and get new insights. We obtained the following results.
・ We obtained a new degree-sum condition for a bipartete graph to have a 2-factor containing a specified edge.
・ We obtained a deep insight on the structure of Ryjacek closure for claw-free graphs. In particular, we proved that if the Ryjacek closure of a claw-free graph G of order n is a complete graph, then G has a cycle of length n - 1.
・ We obtained a degree-sum condition for a graph G to satisfy diff(G) 【less than or equal】 1, where diff(G) is the difference between the order of a longest path and the order of a longest cycle in G.
・ We obtained a new degree-sum condition for a k-factor-critical graph to be hamiltonian.
・ By extending the above result, we obtained a new degree-sum condition for a k-extendable graph to be hamiltonian.
・ We gave a new necessary and sufficient condition for a graph to be 1-extendable in terms of alternating paths. Using this result, we gave a new algorithm to check 1-extendability of a graph G, which runs in O(|E(G)|^2).
・ For given positive integers k and n, we determined the class of graphs such that every cycle has length k (mod n). We proved that this class has a rich structure only if k=0 and n=2, i.e. the class of bipartite graphs.
・ We proved that the Ryjacek closure can be a useful tool in the study of k-factor-critical graphs.
・ We obtained a new degree-sum condition for a bipartite graph to have a 2-factor containing specified vertices in each component. Although the problem is similar to the one at the top of this list, the conclusion was entirely different.
As we see above, we obtained a number of interesting results, and hence the research was successful.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] G.Chen: "Vertex-disjoint cycles containing specified edges in a bipartite graph"Australasian Journal of Combinatorics. 23. 37-48 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Z.Ryjacek: "Claw-free graphs with complete closure"Discrete Mathematics. 236. 325-338 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] R.Li: "Relative length of longest paths and cycles in 3-connected graphs"Journal of Graph Theory. 37. 137-156 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kawarabayashi: "Hamiltonian cycles in n-factor-critical graphs"Discrete Mathematics. 240. 71-82 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kawarabayashi: "Hamiltonian cycles in n-extendable graphs"Journal of Graph Theory. 40. 75-82 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] G.Chen: "The hamiltonicity of bipartite graphs involving neighborhood unions"Discrete Mathematics. 249. 45-56 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] G. Chen: "Vertex-disjoint cycles containing specified edges in a bipartite graph"Australasian J. Combinatorics. 23. 37-48 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Z. Ryjacek: "Claw-free graphs with complete closure"Discrete Math.. 236. 325-338 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] R. Li: "Relative length of longest paths and cycles in 3-connected graphs"J. Graph Theory. 37. 137-156 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Kawarabayashi: "Hamiltonian cycles in n-factor-critical graphs"Discrete Mathematics. 240. 71-82 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Kawarabayashi: "Hamiltonian cycles in n-extendable graphs"J. Graph Theory. 40. 75-82 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] G. Chen: "The hamiltonicity of bipartite graphs involving neighborhood unions"Discrete Mathematics. 249. 45-56 (2002)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2004-04-14  

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