Researches on polychromatic coloring of graphs on surfaces by local transformations

• Project/Area Number: 24540117
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Yokohama National University
• Principal Investigator: NAKAMOTO Atsuhiro 横浜国立大学, 環境情報研究科(研究院), 准教授 (20314445)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000); Fiscal Year 2014: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2012: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
• Keywords: 局所変形 / 曲面 / グラフ彩色 / 三角形分割 / 四角形分割 / 偶三角形分割 / ホモロジー / 偶角形分割 / 多色彩色 / グラフ

Outline of Final Research Achievements

In this research, we dealt with polychromatic coloring of graphs on surfaces. This notion was defined in relation with the so called ``Art gallery problem", and my research purpose is to firstly solve the polychromatic 4-chromaticity of cubic bipartite plane graphs by using local transformations, and secondly to solve the similar problems for even-sided maps on other surfaces as its analogy. As a result, we could solve all problems given in the application as we expected. On the other hand, we found a relation of our result to the extension problem of graphs, and got a new direction for the research to use an algebraic method such as a Z2-homology of a surface and the cycle space of graphs. We have been able to obtain much more than we expected before.

Research Products

• [Journal Article] Quadrangulations on 3-colored point sets with Steiner points and their winding number (2014)
  • Author(s): Sho Kato, Ryuich Mori, Atsuhiro Nakamoto
  • Journal Title: Graphs and Combinatorics Volume: 30 Issue: 5 Pages: 1193-1205
  • DOI: 10.1007/s00373-013-1346-4
  • Peer Reviewed
• [Journal Article] Generating 3-connected quadrangulations on surfaces (2014)
  • Author(s): Momoko Nagashima, A. Nakamoto,Seiya Negami, Yusuke Suzuki
  • Journal Title: Ars Combinatoria Volume: 116
  • Peer Reviewed
• [Journal Article] 3-Polychromatic quadrangulations on surfaces (2014)
  • Author(s): R. Mukae, A. Nakamoto, Y. Suzuki
  • Journal Title: Congressus. Numerantium Volume: 219
  • Peer Reviewed
• [Journal Article] Hamiltonian cycles in bipartite toroidal graphs with a partite set of degree four vertices (2013)
  • Author(s): J. Fujisawa, A. Nakamoto, K. Ozeki
  • Journal Title: J. Combin. Theory Ser. B Volume: 103 Issue: 1 Pages: 46-60
  • DOI: 10.1016/j.jctb.2012.08.004
  • Peer Reviewed
• [Journal Article] HISTs in triangulations on surfaces (2013)
  • Author(s): A. Nakamoto and S. Tsuchiya
  • Journal Title: AKCE Int. J. Graphs Combin. Volume: 10
• [Journal Article] Polychromatic 4-coloring of cubic even embeddings on the projective plane (2013)
  • Author(s): M. Kobayashi, A. Nakamoto and T. Yamaguchi
  • Journal Title: Discrete Math. Volume: 313 Issue: 21 Pages: 2423-2431
  • DOI: 10.1016/j.disc.2013.07.005
  • Peer Reviewed
• [Journal Article] Colored quadrangulation with Steiner points (2013)
  • Author(s): V. Alvarez and A. Nakamoto
  • Journal Title: Lecture Note on Computer Science Volume: 8296 Pages: 20-29
  • DOI: 10.1007/978-3-642-45281-9_2
  • ISBN: 9783642452802, 9783642452819
  • Peer Reviewed
• [Journal Article] The number of diagonal transformations in quadrangulations on the sphere (2013)
  • Author(s): Naoki Matsumoto and Atsuhiro Nakamoto
  • Journal Title: Lecture Notes in Computer Science Volume: 8296 Pages: 110-119
  • DOI: 10.1007/978-3-642-45281-9_11
  • ISBN: 9783642452802, 9783642452819
  • Peer Reviewed
• [Journal Article] Irreducible triangulations of surfaces with boundary (2013)
  • Author(s): A. Boulch, E. Colin de Verdiere and A. Nakamoto
  • Journal Title: Graphs Combin. Volume: 29 Issue: 6 Pages: 1675-1688
  • DOI: 10.1007/s00373-012-1244-1
  • Peer Reviewed
• [Journal Article] Book embedding of toroidal bipartite graphs (2012)
  • Author(s): A. Nakamoto, K. Ota, K. Ozeki
  • Journal Title: SIAM J. Discrete Math. Volume: 26 Issue: 2 Pages: 661-669
  • DOI: 10.1137/100794651
  • Peer Reviewed
• [Journal Article] Coloring Eulerian triangulations of the Klein bottle (2012)
  • Author(s): D. Kral, B. Mohar, A. Nakamoto, O. Pangrac, Y. Suzuki
  • Journal Title: Graphs Combin. Volume: 28 Issue: 4 Pages: 499-530
  • DOI: 10.1007/s00373-011-1063-9
  • Peer Reviewed
• [Journal Article] On geometrically realizable Mobius triangulations (2012)
  • Author(s): A. Nakamoto, S. Tsuchiya
  • Journal Title: Discrete Math. Volume: 312 Issue: 14 Pages: 2135-2139
  • DOI: 10.1016/j.disc.2011.06.007
• [Journal Article] Y-rotation in k-minimal quadrangulations on the projective plane (2012)
  • Author(s): 鈴木有祐, 中本敦浩
  • Journal Title: Journal of Graph Theory Volume: 69 Issue: 3 Pages: 301-313
  • DOI: 10.1002/jgt.20583
  • Peer Reviewed
• [Journal Article] A face of a projective triangulation removed for its geometric realizability (2012)
  • Author(s): A. Nakamoto, S. Tsuchiya
  • Journal Title: Discrete Comput. Geom. Volume: 47 Issue: 1 Pages: 215-234
  • DOI: 10.1007/s00454-011-9374-4
• [Journal Article] Polychromatic 4-coloring of cubic bipartite plane graphs (2012)
  • Author(s): E. Horev, M.J. Katz, R. Krakovski, A. Nakamoto
  • Journal Title: Discrete Math. Volume: 312 Issue: 4 Pages: 715-719
  • DOI: 10.1016/j.disc.2011.11.016
• [Journal Article] Hamiltonian cycles in bipartite quadrangulations on the torus (2012)
  • Author(s): A. Nakamoto, K. Ozeki
  • Journal Title: J. Graph Theory Volume: 69 Issue: 2 Pages: 143-151
  • DOI: 10.1002/jgt.20569
• [Presentation] Book embedding of graphs on surfaces (2015)
  • Author(s): Atsuhiro Nakamoto
  • Organizer: Forty-sixth Southeastern International Conferenceon Combinatorics, Graph Theory, and Computing
  • Place of Presentation: Florida Atlantic University, USA
  • Year and Date: 2015-03-02 – 2015-03-06
• [Presentation] Book embedding of graphs embeddable on surfaces (2014)
  • Author(s): Atsuhiro Nakamoto
  • Organizer: ACOTTA2014
  • Place of Presentation: Ixtapa, Mexico
  • Year and Date: 2014-11-24 – 2014-11-28
• [Presentation] Odd minors in even embeddings on surfaces (2014)
  • Author(s): 中本敦浩
  • Organizer: 第24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: 2014-11-07 – 2014-11-08
• [Presentation] Odd minors in even embeddings on surfaces (2014)
  • Author(s): 中本敦浩
  • Organizer: 離散数学とその応用研究集会
  • Place of Presentation: 新潟総合テレビ・ゆめディア
  • Year and Date: 2014-08-20 – 2014-08-22
• [Presentation] Coloring maps on surfaces (2014)
  • Author(s): Atsuhiro Nakamoto
  • Organizer: SIAM Conference on Discrete Mathematics
  • Place of Presentation: Minneapolis, USA
  • Year and Date: 2014-06-16 – 2014-06-19
  • Invited
• [Presentation] 3-Polychromatic quadrangulations on surfaces (2014)
  • Author(s): A. Nakamoto
  • Organizer: 45th Southeastern International Conference on Combinatorics, Graph Theory, and Computing
  • Place of Presentation: Florida Atlantic大学（アメリカ）
• [Presentation] Polychromatic number of graphs on surfaces (2014)
  • Author(s): A. Nakamoto
  • Organizer: The 3rd Taiwan-Japan Conference on Combinatorics and its Applications
  • Place of Presentation: National Chiayi University（台湾）
  • Invited
• [Presentation] Generating Eulerian triangulations on surfaces (2013)
  • Author(s): A. Nakamoto
  • Organizer: Geometrical and Topological Graph Theory
  • Place of Presentation: Bristol大学（イギリス）
• [Presentation] Grunbaum coloring of Eulerain triangulations on surfaces (2013)
  • Author(s): A. Nakamoto
  • Organizer: 7th Czech-Slovak International Symposium on Graph Theory, Combinatorics, Algorithms and Applications
  • Place of Presentation: Kosice（スロバキア）
• [Presentation] Odd minors and cycle parities of even embeddings on surfaces (2012)
  • Author(s): Atsuhiro Nakamoto
  • Organizer: International Conference on Cycles in Graphs
  • Place of Presentation: Vanderbilt University (USA)
• [Presentation] Polychromatic coloring of graphs on surfaces (2012)
  • Author(s): Atsuhiro Nakamoto
  • Organizer: SIAM Conference on Discrete Mathematics
  • Place of Presentation: Dalhousie University (CANADA)
• [Presentation] Colored Quadrangulations with Steiner Points (2012)
  • Author(s): Atsuhiro Nakamoto
  • Organizer: Thailand-. Japan Joint Conference on Computational Geometry and Graphs 2012
  • Place of Presentation: Srinakharinwirot University (THAILAND)