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Construction of single-path designs and 2-path designs

Research Project

Project/Area Number 19540142
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 InstitutionUniversity of Shizuoka

Principal Investigator

KOBAYASHI Midori (MIDORI Kobayashi)  University of Shizuoka, 経営情報学部, 教授 (00136631)

Project Period (FY) 2007 – 2010
Project Status Completed (Fiscal Year 2010)
Budget Amount *help
¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2009: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2008: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2007: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
KeywordsDudeney集合 / 2パスデザイン / グラフ理論 / デザイン / サイクル / パス
Research Abstract

A set of Hamilton cycles in a graph is called a Dudeney set if every 2-path lies on exactly one of the cycles. It has been conjectured that there is a Dudeney set in the complete graph of order n for all n. It is known that there exists a Dudeney set when n is even, but the conjecture is still unsettled when n is odd.
In this reseach, we defined a black 1-factor, and proved that if there exists a black 1-factor then we can construct a Dudeney set. We also proved that if there is a black 1-factor of order p+1, then 2 is a quadratic residue modulo p. Using this result, we obtained some new Dudeney sets.
We studied the existence problem of a Dudeney set in the complete bipartite graph of order 2n and we proved the following theorem. There exists a Dudeney set in the complete bipartite graph of order 2n when n=0,1,3 (mod 4), and there exists a double Dudeney set when n=2 (mod 4). When n=2 (mod 4), the existence problem of a Dudeney set remains open.
We studied the problem on a Dudeney set of 4-cycles, and we proved that there exists a Dudeney set of 4-cycles if and only if one of the following holds: (i) n is even, or (ii) n is odd and λis even.
We have studied graceful labelings of trees and have found new graceful labelings of some trees. We also obtained some results on construction of some combinatorial designs.

Report

(6 results)
  • 2010 Annual Research Report   Final Research Report ( PDF )
  • 2009 Annual Research Report   Self-evaluation Report ( PDF )
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • Research Products

    (36 results)

All 2010 2009 2008 Other

All Journal Article (22 results) (of which Peer Reviewed: 10 results) Presentation (14 results)

  • [Journal Article] Black 1-factors and Dudeney sets2010

    • Author(s)
      M. Kobayashi, B. McKay, N. Mutoh, G. Nakamura
    • Journal Title

      Journal of Combinatorial Mathematics and Combinatorial Computing Vol.75

      Pages: 167-174

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Exact coverings of 2-paths with Hamilton cycles for the complete bipartite graph2010

    • Author(s)
      M. Kobayashi, G. Nakamura
    • Journal Title

      Working paper series, School of Administration and Informatics, University of Shizuoka

      Pages: 1-5

    • Related Report
      2010 Final Research Report
  • [Journal Article] Uniform coverings of 2-paths with 4-cycles in the complete bipartite graph2010

    • Author(s)
      M. Kobayashi, G. Nakamura
    • Journal Title

      Working paper series, School of Administration and Informatics, University of Shizuoka

      Pages: 1-4

    • Related Report
      2010 Final Research Report
  • [Journal Article] Graceful Labeling of some trees2010

    • Author(s)
      M. Kobayashi, C. Nara, G. Nakamura
    • Journal Title

      Working paper series 2010, School of Administration and Informatics, University of Shizuoka

      Pages: 1-10

    • Related Report
      2010 Final Research Report
  • [Journal Article] Black 1-factors and Dudeney sets2010

    • Author(s)
      M.Kobayashi, B.McKay, N.Mutoh, G.Nakamura
    • Journal Title

      Journal of Combinatorial Mathematics and Combinatorial Computing

      Volume: 75 Pages: 167-174

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Graceful Labeling of some trees2010

    • Author(s)
      M.Kobayashi, C.Nara, G.Nakamura
    • Journal Title

      Working paper series, School of Administration and Informatics, University of Shizuoka

      Pages: 1-10

    • Related Report
      2010 Annual Research Report
  • [Journal Article] Uniform coverings of 2-paths with 4-cycles in the complete bipartite graph2010

    • Author(s)
      M.Kobayashi, G.Nakamura
    • Journal Title

      Working paper series, School of Administration and Informatics, University of Shizuoka

      Pages: 1-4

    • Related Report
      2010 Annual Research Report
  • [Journal Article] Graceful Labeling of Susuki trees, Working paper series2010

    • Author(s)
      M. Kobayashi, C. Nara, G. Nakamura
    • Journal Title

      School of Administration and Informatics(University of Shizuoka) 1001

      Pages: 1-5

    • Related Report
      2009 Self-evaluation Report
  • [Journal Article] Graceful Labeling of Susuki trees2010

    • Author(s)
      M.Kobayashi, C.Nara, G.Nakamura
    • Journal Title

      Working paper series 1001, School of Administration and Informatics, University of Shizuoka

      Pages: 1-5

    • Related Report
      2009 Annual Research Report
  • [Journal Article] Interlaced Graceful Labeling of Firecrackers2009

    • Author(s)
      M. Kobayashi, C. Nara, G. Nakamura
    • Journal Title

      Working paper series 0902, School of Administration and Informatics, University of Shizuoka

      Pages: 1-5

    • Related Report
      2010 Final Research Report 2008 Annual Research Report
  • [Journal Article] Interlaced Graceful Labeling of Firecrackers, Working paper series2009

    • Author(s)
      M. Kobayashi, C. Nara, G. Nakamura
    • Journal Title

      School of Administration and Informatics(University of Shizuoka) 0902

      Pages: 1-5

    • Related Report
      2009 Self-evaluation Report
  • [Journal Article] Black 1-factors and Dudeney sets, revised version2009

    • Author(s)
      M. Kobayashi, B. McKay, N. Mutoh, G. Nakamura
    • Journal Title

      Working paper series 0901, School of Administration and Informatics, University of Shizuoka

      Pages: 1-6

    • Related Report
      2008 Annual Research Report
  • [Journal Article] アダマール行列の一般化とその応用2009

    • Author(s)
      小林, 宮内, 武藤, 中村
    • Journal Title

      経営と情報 第21巻

      Pages: 15-26

    • NAID

      110007172466

    • Related Report
      2008 Annual Research Report
  • [Journal Article] Dudeney transformation of normal tiles, Lecture Notes in Computer Science2008

    • Author(s)
      J. Akiyama, M. Kobayashi, G. Nakamura
    • Journal Title

      Springer 4535

      Pages: 10-13

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Semi-Distance Codes and Steiner Systems2008

    • Author(s)
      H. Ito, M. Kobayashi, G. Nakamura
    • Journal Title

      Graphs and Combinatorics 23

      Pages: 283-290

    • Related Report
      2010 Final Research Report 2009 Self-evaluation Report 2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Dudeney transformation of normal tiles2008

    • Author(s)
      J. Akiyama, M. Kobayashi, G. Nakamura
    • Journal Title

      Lecture Notes in Computer Science(Springer) 4535

      Pages: 10-13

    • Related Report
      2009 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Dudeney transformation of normal tiles2008

    • Author(s)
      J. Akiyama, M. Kobayashi, G. Nakamura
    • Journal Title

      Lecture Notes in Computer Science, Springer 4535

      Pages: 1-13

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Black 1-factors and Dudeney sets2008

    • Author(s)
      M. Kobayashi, B. McKay, N. Mutoh, G. Nakamura
    • Journal Title

      Working paper series 0801, School of Administration and Informatics, University of Shizuoka.

      Pages: 1-7

    • Related Report
      2007 Annual Research Report
  • [Journal Article] Arrangements of n points whose incident-line-numbers are at most n/2 the JCCGG2009 special issue of Graphs and Combinatorics (G&C)

    • Author(s)
      J. Akiyama, H. Ito, M. Kobayashi, G. Nakamura
    • Journal Title

      Springer, accepted

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Black 1-factors and Dudeney sets

    • Author(s)
      M. Kobayashi, B. McKay, N. Mutoh, G. Nakamura
    • Journal Title

      Ars Combinatoria (印刷中)

    • Related Report
      2009 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Black 1-factors and Dudeney sets

    • Author(s)
      M.Kobayashi, B.McKay, N.Mutoh, G.Nakamura
    • Journal Title

      Ars Combinatoria (印刷中)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Dudeney transformation of normal tiles

    • Author(s)
      J. Akiyama, M. Kobayashi, G. Nakamura
    • Journal Title

      Lecture Notes in Computer Science, Springer (印刷中)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] Magic squares with powered sum the China-Japan Joint Conference on Computational Geometry2010

    • Author(s)
      S. Chikaraishi, M. Kobayashi, N. Mutoh, G. Nakamura
    • Organizer
      Graphs and Applications
    • Place of Presentation
      大連, 中国
    • Year and Date
      2010-11-06
    • Related Report
      2010 Final Research Report
  • [Presentation] Magic squares with powered sum2010

    • Author(s)
      S.Chikaraishi, M.Kobayashi, N.Mutoh, G.Nakamura
    • Organizer
      the China-Japan Joint Conference on Computational Geometry, Graphs and Applications
    • Place of Presentation
      大連,中国
    • Year and Date
      2010-11-06
    • Related Report
      2010 Annual Research Report
  • [Presentation] Exact coverings of 2-paths with 4-cycles in the complete, bipartite graph2010

    • Author(s)
      M. Kobayashi, G. Nakamura
    • Organizer
      the 54th Annual Meeting of the Australian Mathematical Society
    • Place of Presentation
      Brisbane, Australia
    • Year and Date
      2010-09-29
    • Related Report
      2010 Final Research Report
  • [Presentation] Exact coverings of 2-paths with 4-cycles in the complete bipartite graph2010

    • Author(s)
      M.Kobayashi, G.Nakamura
    • Organizer
      the 54th Annual Meeting of the Australian Mathematical Society
    • Place of Presentation
      Brisbane, Australia
    • Year and Date
      2010-09-29
    • Related Report
      2010 Annual Research Report
  • [Presentation] Arrangements of n points whose incident-line-numbers are at most n/22009

    • Author(s)
      H.Ito, M.Kobayashi, G.Nakamura
    • Organizer
      Japan Conference on Computational Geometry and Graphs
    • Place of Presentation
      金沢市
    • Year and Date
      2009-11-11
    • Related Report
      2009 Annual Research Report
  • [Presentation] Arrangements of eleven points in the plane, each with five incident lines2009

    • Author(s)
      H.Ito, M.Kobayashi, G.Nakamura
    • Organizer
      Japan Conference on Computational Geometry and Graphs
    • Place of Presentation
      金沢市
    • Year and Date
      2009-11-11
    • Related Report
      2009 Annual Research Report
  • [Presentation] Arrangements of n points whose incident-line-numbers are at most n/22009

    • Author(s)
      H. Ito, M. Kobayashi, G. Nakamura
    • Organizer
      Japan Conference on Computational Geometry and Graphs
    • Place of Presentation
      Kanazawa, Japan
    • Related Report
      2010 Final Research Report 2009 Self-evaluation Report
  • [Presentation] Arrangements of eleven points in the plane, each with five incident lines2009

    • Author(s)
      H. Ito, M. Kobayashi, G. Nakamura
    • Organizer
      Japan Conference on Computational Geometry and Graphs
    • Place of Presentation
      Kanazawa, Japan
    • Related Report
      2010 Final Research Report
  • [Presentation] Nakamura, Arrangements of eleven points in the plane, each with five incident lines2009

    • Author(s)
      H. Ito, M. Kobayashi
    • Organizer
      Japan Conference on Computational Geometry and Graphs
    • Place of Presentation
      Kanazawa, Japan
    • Related Report
      2009 Self-evaluation Report
  • [Presentation] Dudeney transformation of normal tiles2008

    • Author(s)
      J. Akiyama, M. Kobayashi, G. Nakamura
    • Organizer
      The Kyoto International Conference on Computational Geometry and Graph Theory
    • Place of Presentation
      京都大学
    • Year and Date
      2008-06-14
    • Related Report
      2009 Self-evaluation Report 2007 Annual Research Report
  • [Presentation] Dudeney transformation of normal tiles2008

    • Author(s)
      J. Akiyama, M. Kobayashi, G. Nakamura
    • Organizer
      General topologyシンポジウム
    • Place of Presentation
      高崎
    • Related Report
      2010 Final Research Report
  • [Presentation] Dudeney transformation of normal tiles2008

    • Author(s)
      Akiyama, M. Kobayashi, G. Nakamura
    • Organizer
      General topology シンポジウム
    • Place of Presentation
      高崎経済大学
    • Related Report
      2009 Self-evaluation Report 2008 Annual Research Report
  • [Presentation] アダマール行列の一般化とその応用2008

    • Author(s)
      小林みどり, 宮内美樹, 武藤伸明, 中村義作
    • Organizer
      離散数学の統計科学および関連分野への応用研究集会
    • Place of Presentation
      岐阜県下呂市
    • Related Report
      2009 Self-evaluation Report
  • [Presentation] アダマール行列の一般化2008

    • Author(s)
      小林, 宮内, 中村
    • Organizer
      離散数学の統計科学および関連分野への応用研究集会
    • Place of Presentation
      岐阜県下呂市
    • Related Report
      2008 Annual Research Report

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Published: 2007-04-01   Modified: 2016-04-21  

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