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

• Project/Area Number: 18740045
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Yokohama 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: ¥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。

Research Products

• [Journal Article] 3-trees with a few vertices of degree 3 in 3-connected planar graphs (2009)
  • Author(s): A. Nakamoto, K. Ota and Y. Oda
  • Journal Title: Discrete Math. 309 Pages: 666-672
  • Peer Reviewed
• [Journal Article] K6-Minors in triangulations and complete quadrangulations (2009)
  • Author(s): R. Mukae and A. Nakamoto
  • Journal Title: J. Graph Theory 60 Pages: 302-312
  • Peer Reviewed
• [Journal Article] N-Flips in even triangulations on surfaces (2009)
  • Author(s): K. Kawarabayashi, A. Nakamoto and Y. Suzuki
  • Journal Title: J. Combin. Theory, Ser. B 99 Pages: 229-246
  • Peer Reviewed
• [Journal Article] N-Flips in even triangulations on surfaces (2009)
  • Author(s): K. Kawarabayashi
  • Journal Title: Journal of Combinatorial Theory, Series B 99 Pages: 229-246
  • Peer Reviewed
• [Journal Article] Geometric realization of triangulations on the Mobius band (2008)
  • Author(s): M. Chavez, G. Fijavz, A Marquez, A. Nakamoto and E. Suarez
  • Journal Title: SIAM J. Discrete Math. 23 Pages: 221-232
  • Peer Reviewed
• [Journal Article] K6-Minors in triangulations in triangulations on the Klein bottle (2008)
  • Author(s): K. Kawarabayashi, R. Mukae and A. Nakamoto
  • Journal Title: SIAM J. Discrete Math. 23 Pages: 96-108
  • Peer Reviewed
• [Journal Article] N-Flips in even triangulations on the projective plane (2008)
  • Author(s): A. Nakamoto and Y. Suzuki
  • Journal Title: Discrete Math. 308 Pages: 5454-5462
  • Peer Reviewed
• [Journal Article] Geometric realization of a projective triangulation with one face removed (2008)
  • Author(s): P.C. Bonnington, A. Nakamoto
  • Journal Title: Discrete Comp. Geom. 40 Pages: 141-157
  • Peer Reviewed
• [Journal Article] 5-Chromatic even train-gulations on surfaces (2008)
  • Author(s): A. Nakamoto
  • Journal Title: Discrete Math. 308 Pages: 2571-2580
  • Peer Reviewed
• [Journal Article] 5-Chromatic even triangulations on surfaces (2008)
  • Author(s): A. Nakamoto
  • Journal Title: Discrete Mathematics 307 Pages: 2571-2580
  • Peer Reviewed
• [Journal Article] Geometric realization of a projective triangulation with one face removed (2008)
  • Author(s): P. Bonnington
  • Journal Title: Discrete and Computational Geometry 40 Pages: 141-157
  • Peer Reviewed
• [Journal Article] The Erdos - Poca Property for vertex and edge-disjoint odd cycles in graphs on orientable surfaces (2007)
  • Author(s): K. Kawarabayashi and A. Nakamoto
  • Journal Title: Discrete Math. 307 Pages: 764-768
  • Peer Reviewed
• [Journal Article] 3-Coloring of even embeddings on surfaces (2007)
  • Author(s): A. Nakamoto and N. Sasanuma
  • Journal Title: Graphs Combin. 23 Pages: 87-95
  • Peer Reviewed
• [Journal Article] K6-Minors in triangulations on the double torus (2007)
  • Author(s): A. Nakamoto, Y. Oda and K, Ota
  • Journal Title: Congr. Numer. 188 Pages: 150-160
  • Peer Reviewed
• [Journal Article] 3-Colorable even embeddings on closed surfaces (2007)
  • Author(s): A.Nakamoto
  • Journal Title: Graphs and Combinatrics 23 Pages: 78-95
  • Peer Reviewed
• [Journal Article] The Erdos-Poca property for vertex-and edge-disjoint odd cycles in graphs on orientable surfaces (2007)
  • Author(s): A.Nakamoto
  • Journal Title: Discrete Mathematics 307 Pages: 764-768
  • Peer Reviewed
• [Journal Article] Tight quadrangulations on the sphere (2006)
  • Author(s): K. Ando, H. Komuro and A. Nakamoto
  • Journal Title: Discrete Math. 360 Pages: 278-283
  • Peer Reviewed
• [Journal Article] N-Flips in even triangulations on the sphere (2006)
  • Author(s): A. Nakamoto, T. Sakuma and Y. Suzuki
  • Journal Title: J. Graph Theory 51 Pages: 260-268
  • Peer Reviewed
• [Journal Article] Tight quadrangulations on the sphere (2006)
  • Author(s): K.Ando, H.Komuro, A.Nakamoto
  • Journal Title: Discrete Mathematics 360 Pages: 278-283
• [Journal Article] N-Flips in even triangulations on the sphere (2006)
  • Author(s): A.Nakamoto, T.Sakuma, Y.Suzuki
  • Journal Title: Journal of Graph Theory 51 Pages: 260-268
• [Journal Article] Dominating sets in traingulations on surfaces
  • Author(s): K. Kawarabayashi, T. Honjo and A. Nakamoto
  • Journal Title: J. Graph Theory (in-press)
  • Peer Reviewed
• [Presentation] Dominating sets in triangulations on surfaces (2008)
  • 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
• [Presentation] Dominating sets in triangulations on surfaces (2008)
  • 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
• [Presentation] Geometric realization of a triangulation on the projective plane with one face removed (2008)
  • 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
• [Presentation] Geometric realization of a triangulation on the projective plane with one face removed (2008)
  • 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
• [Presentation] Dominating sets in triangulations on surfaces (2008)
  • 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
• [Presentation] Geometric realization of a triangulation on the projective plane with one face removed (2007)
  • 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
• [Presentation] N-Flips in even triangulations and Dehn twists on the torus (2007)
  • 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
• [Presentation] グラフの変形と曲面のDehn twists (2007)
  • 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
• [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
• [Presentation] K6-Minors in triangulations on the double torus (2007)
  • 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
• [Presentation] Geometric realization of a triangulation on the projective plane with one face removed (2007)
  • 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
• [Presentation] Geometric realization of a triangulation on the projective plane with one face removed (2007)
  • 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
• [Presentation] K6-Minors in triangulations on surfaces (2007)
  • 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
• [Presentation] Chromatic number of even triangulations on the Klein bottle (2006)
  • 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
• [Presentation] Geometric realization of a 5-connected triangulation on the Mobius band (2006)
  • 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
• [Presentation] Geometric realization of a 5-connected triangulation on the Mebius band (2006)
  • 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
• [Presentation] 5-Chromatic even triangulations on surfaces (2006)
  • 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
• [Remarks] ホームページ
  • URL: http://www.ms.edhs.ynu.ac.jp/Nakamoto-lab/