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A study on polyhedral realization of graphs on surfaces using graph minor theory and local deformations

Research Project

Project/Area Number 18740045
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionYokohama National University

Principal Investigator

ATSUHIRO Nakamoto  Yokohama National University, 教育人間科学部, 准教授 (20314445)

Project Period (FY) 2006 – 2008
Project Status Completed (Fiscal Year 2008)
Budget Amount *help
¥3,960,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥360,000)
Fiscal Year 2008: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2007: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2006: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywords曲面 / 射影平面 / メビウスの帯 / グラフ / 三角形分割 / 多面体 / グラフマイナー / 局所変形 / 多面体的実現 / representativity / 連結度
Research Abstract

曲面上に与えられたグラフが、3次元ユークリッド空間に多面体として実現可能か(多面体的実現可能か)という問題の歴史は古く、最も古い研究は1922年のSteinitzの定理にさかのぼるだろう。1960年代にGrumbaumが「任意の向き付け可能な閉曲面の三角形分割は多面体的実現可能である」ことを予想したが、Steinitzの球面上の定理が証明されて以来、それは長い間未解決であった。しかし、2006年、Bokowskiたちは種数6の曲面に反例を見つけ、2007年にはArchdeaconたちがトーラス上ではこの予想が正しいことを証明した。本研究では、それらの研究の流れを汲み、向き付け不可能な種数1の閉曲面(射影平面)上の三角形分割から1つの面を取り除いて得られるメビウスの帯の三角形分割の多面体的実現の研究を行った。特に、射影平面の任意の三角形分割は、ある1つの面を取り除くと、多面体的実現を持つことを証明し、さらに、任意の1つの面を取り除くことにより多面体的実現を持つ三角形分割を特徴付けた。

Report

(4 results)
  • 2008 Annual Research Report   Final Research Report ( PDF )
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • Research Products

    (39 results)

All 2009 2008 2007 2006 Other

All Journal Article (21 results) (of which Peer Reviewed: 19 results) Presentation (17 results) Remarks (1 results)

  • [Journal Article] 3-trees with a few vertices of degree 3 in 3-connected planar graphs2009

    • Author(s)
      A. Nakamoto, K. Ota and Y. Oda
    • Journal Title

      Discrete Math. 309

      Pages: 666-672

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] K6-Minors in triangulations and complete quadrangulations2009

    • Author(s)
      R. Mukae and A. Nakamoto
    • Journal Title

      J. Graph Theory 60

      Pages: 302-312

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] N-Flips in even triangulations on surfaces2009

    • Author(s)
      K. Kawarabayashi, A. Nakamoto and Y. Suzuki
    • Journal Title

      J. Combin. Theory, Ser. B 99

      Pages: 229-246

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] N-Flips in even triangulations on surfaces2009

    • Author(s)
      K. Kawarabayashi
    • Journal Title

      Journal of Combinatorial Theory, Series B 99

      Pages: 229-246

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Geometric realization of triangulations on the Mobius band2008

    • Author(s)
      M. Chavez, G. Fijavz, A Marquez, A. Nakamoto and E. Suarez
    • Journal Title

      SIAM J. Discrete Math. 23

      Pages: 221-232

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] K6-Minors in triangulations in triangulations on the Klein bottle2008

    • Author(s)
      K. Kawarabayashi, R. Mukae and A. Nakamoto
    • Journal Title

      SIAM J. Discrete Math. 23

      Pages: 96-108

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] N-Flips in even triangulations on the projective plane2008

    • Author(s)
      A. Nakamoto and Y. Suzuki
    • Journal Title

      Discrete Math. 308

      Pages: 5454-5462

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Geometric realization of a projective triangulation with one face removed2008

    • Author(s)
      P.C. Bonnington, A. Nakamoto
    • Journal Title

      Discrete Comp. Geom. 40

      Pages: 141-157

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] 5-Chromatic even train-gulations on surfaces2008

    • Author(s)
      A. Nakamoto
    • Journal Title

      Discrete Math. 308

      Pages: 2571-2580

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] 5-Chromatic even triangulations on surfaces2008

    • Author(s)
      A. Nakamoto
    • Journal Title

      Discrete Mathematics 307

      Pages: 2571-2580

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Geometric realization of a projective triangulation with one face removed2008

    • Author(s)
      P. Bonnington
    • Journal Title

      Discrete and Computational Geometry 40

      Pages: 141-157

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Erdos - Poca Property for vertex and edge-disjoint odd cycles in graphs on orientable surfaces2007

    • Author(s)
      K. Kawarabayashi and A. Nakamoto
    • Journal Title

      Discrete Math. 307

      Pages: 764-768

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] 3-Coloring of even embeddings on surfaces2007

    • Author(s)
      A. Nakamoto and N. Sasanuma
    • Journal Title

      Graphs Combin. 23

      Pages: 87-95

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] K6-Minors in triangulations on the double torus2007

    • Author(s)
      A. Nakamoto, Y. Oda and K, Ota
    • Journal Title

      Congr. Numer. 188

      Pages: 150-160

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] 3-Colorable even embeddings on closed surfaces2007

    • Author(s)
      A.Nakamoto
    • Journal Title

      Graphs and Combinatrics 23

      Pages: 78-95

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Erdos-Poca property for vertex-and edge-disjoint odd cycles in graphs on orientable surfaces2007

    • Author(s)
      A.Nakamoto
    • Journal Title

      Discrete Mathematics 307

      Pages: 764-768

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Tight quadrangulations on the sphere2006

    • Author(s)
      K. Ando, H. Komuro and A. Nakamoto
    • Journal Title

      Discrete Math. 360

      Pages: 278-283

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] N-Flips in even triangulations on the sphere2006

    • Author(s)
      A. Nakamoto, T. Sakuma and Y. Suzuki
    • Journal Title

      J. Graph Theory 51

      Pages: 260-268

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Tight quadrangulations on the sphere2006

    • Author(s)
      K.Ando, H.Komuro, A.Nakamoto
    • Journal Title

      Discrete Mathematics 360

      Pages: 278-283

    • Related Report
      2006 Annual Research Report
  • [Journal Article] N-Flips in even triangulations on the sphere2006

    • Author(s)
      A.Nakamoto, T.Sakuma, Y.Suzuki
    • Journal Title

      Journal of Graph Theory 51

      Pages: 260-268

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Dominating sets in traingulations on surfaces

    • Author(s)
      K. Kawarabayashi, T. Honjo and A. Nakamoto
    • Journal Title

      J. Graph Theory (in-press)

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Presentation] Dominating sets in triangulations on surfaces2008

    • Author(s)
      中本敦浩
    • Organizer
      Workshop on Topological Graph Theory in Yokohama (TGT20)
    • Place of Presentation
      Yokohama National University
    • Year and Date
      2008-11-24
    • Related Report
      2008 Final Research Report
  • [Presentation] Dominating sets in triangulations on surfaces2008

    • Author(s)
      中本敦浩
    • Organizer
      純粋数学及び応用数学としての組合せ論・離散幾何そしてその周辺ワークショップ
    • Place of Presentation
      電気通信大学
    • Year and Date
      2008-07-05
    • Related Report
      2008 Final Research Report
  • [Presentation] Geometric realization of a triangulation on the projective plane with one face removed2008

    • Author(s)
      A. Nakamoto
    • Organizer
      Topological and Geometrical Graph Theory
    • Place of Presentation
      パリ高等師範学校パリ,フランス
    • Year and Date
      2008-05-23
    • Related Report
      2008 Annual Research Report
  • [Presentation] Geometric realization of a triangulation on the projective plane with one face removed2008

    • Author(s)
      中本敦浩
    • Organizer
      Topological and Geometrical Graph Theory 2008
    • Place of Presentation
      Ecole Normale Superieure, Paris, France
    • Year and Date
      2008-05-22
    • Related Report
      2008 Final Research Report
  • [Presentation] Dominating sets in triangulations on surfaces2008

    • Author(s)
      中本敦浩
    • Organizer
      The 24th PNU-POSTECH Algebraic Combinatorics Seminar
    • Place of Presentation
      Pusan National University, Korea
    • Year and Date
      2008-03-08
    • Related Report
      2008 Final Research Report
  • [Presentation] Geometric realization of a triangulation on the projective plane with one face removed2007

    • Author(s)
      中本敦浩
    • Organizer
      The 32th Australasian Conference on Combinatorial Mathematics and Combinatorial Computing
    • Place of Presentation
      Otago University, Dunedin, New Zealand
    • Year and Date
      2007-12-03
    • Related Report
      2008 Final Research Report
  • [Presentation] N-Flips in even triangulations and Dehn twists on the torus2007

    • Author(s)
      中本敦浩
    • Organizer
      第19回位相幾何学的グラフ理論研究集会
    • Place of Presentation
      横浜国立大学
    • Year and Date
      2007-11-17
    • Related Report
      2008 Final Research Report
  • [Presentation] グラフの変形と曲面のDehn twists2007

    • Author(s)
      中本敦浩
    • Organizer
      短期共同研究
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-10-03
    • Related Report
      2008 Final Research Report
  • [Presentation] 位相幾何学的グラフ理論「曲面上のグラフの染色数」2007

    • Author(s)
      中本敦浩
    • Organizer
      組合せ論サマースクール
    • Place of Presentation
      カルチャーリゾートフェストーネ
    • Year and Date
      2007-09-04
    • Related Report
      2008 Final Research Report
  • [Presentation] K6-Minors in triangulations on the double torus2007

    • Author(s)
      中本敦浩
    • Organizer
      研究集会「組合せ論と離散幾何」
    • Place of Presentation
      下呂市民会館
    • Year and Date
      2007-08-07
    • Related Report
      2008 Final Research Report
  • [Presentation] Geometric realization of a triangulation on the projective plane with one face removed2007

    • Author(s)
      中本敦浩
    • Organizer
      International Conference on Computational Geometry and Graph Theory
    • Place of Presentation
      Kyoto CGGT2007
    • Year and Date
      2007-06-12
    • Related Report
      2008 Final Research Report
  • [Presentation] Geometric realization of a triangulation on the projective plane with one face removed2007

    • Author(s)
      A. Nakamoto
    • Organizer
      International conference on computational geometry and graph theory
    • Place of Presentation
      京都大学
    • Year and Date
      2007-06-12
    • Related Report
      2007 Annual Research Report
  • [Presentation] K6-Minors in triangulations on surfaces2007

    • Author(s)
      中本敦浩
    • Organizer
      38th South Eastern International Conference on Combinatorics Graph Theory and Computing
    • Place of Presentation
      Florida Atlantic University
    • Year and Date
      2007-03-09
    • Related Report
      2008 Final Research Report
  • [Presentation] Chromatic number of even triangulations on the Klein bottle2006

    • Author(s)
      中本敦浩
    • Organizer
      第18回位相幾何学的グラフ理論研究集会
    • Place of Presentation
      横浜国立大学
    • Year and Date
      2006-11-17
    • Related Report
      2008 Final Research Report
  • [Presentation] Geometric realization of a 5-connected triangulation on the Mobius band2006

    • Author(s)
      中本敦浩
    • Organizer
      日本数学会2006年度会
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2006-09-20
    • Related Report
      2008 Final Research Report
  • [Presentation] Geometric realization of a 5-connected triangulation on the Mebius band2006

    • Author(s)
      中本敦浩
    • Organizer
      研究集会「離散数学とその応用」
    • Place of Presentation
      茨城大学工学部
    • Year and Date
      2006-08-03
    • Related Report
      2008 Final Research Report
  • [Presentation] 5-Chromatic even triangulations on surfaces2006

    • Author(s)
      中本敦浩
    • Organizer
      SIAM conference on Discrete Mathematics
    • Place of Presentation
      University of Victoria, Canada
    • Year and Date
      2006-06-26
    • Related Report
      2008 Final Research Report
  • [Remarks] ホームページ

    • URL

      http://www.ms.edhs.ynu.ac.jp/Nakamoto-lab/

    • Related Report
      2008 Final Research Report

URL: 

Published: 2006-04-01   Modified: 2016-04-21  

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