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2002 Fiscal Year Final Research Report Summary

Research on Topological Aspects of Combinatorics

Research Project

Project/Area Number 13640134
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 InstitutionKeio University

Principal Investigator

OTA Katsuhiro  Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (40213722)

Co-Investigator(Kenkyū-buntansha) JIMBO Masakazu  Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (50103049)
MAEDA Yoshiaki  Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (40101076)
ENOMOTO Hikoe  Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (00011669)
NAKAMOTO Atsuhiro  Yokohama National University, Faculty of Education and Human Sciences, Lecturer, 教育人間科学部, 講師 (20314445)
NEGAMI Seiya  Yokohama National University, Faculty of Education and Human Sciences, Professor, 教育人間科学部, 教授 (40164652)
Project Period (FY) 2001 – 2002
KeywordsCyclic chromatic number / Locally bipartite graph / Locally planar graph / Triangulation / Spanning tree / Topological graph theory
Research Abstract

As a result on combinatorial property of planar graphs, we have proved that if the maximum degree of a planar graph is sufficiently large, then its cyclic chromatic number is at most the maximum degree plus one. On the coloring of locally planar graphs on surfaces, we have found that the orientability of the surface plays an important role. In particular, we gave a topological characterization of the quadrangulations on the torus and the Klein bottle having chromatic number 3 and 4, respectively. For general nonorientable surfaces, we characterized all graphs having chromatic number 5.
There are many researches on triangulations of surfaces. Two triangulations of the same large order can be transformed by a sequence of diagonal transformations. We studied the number of times needed transformation, and proved that it is bounded by a linear function of the order. Also, we obtained some results on transformation of two graphs with the same face size distributions. Related to these researches, we considered the graphs whose edges are all incident with a vertex of degree d. The graph with this property is called d-covered. We gave a constructive characterization of 5-covered and 6-covered triangulations.
For locally planar 3-connected graphs on a surface, using a general method to obtain a spanning planar subgraph with good property, we have obtained several properties of such 3-connected graphs which are close to hamiltonicity. As results improving the known results, we have proved the existence of almost 7-coverings, almost 3-trees, and 4-trees with bounded number of vertices of degree at least 3.
We have also obtained some results on Ramsey theorem on spatial graphs, graph partition problems, reembedding structure of triangulation, and finite planar coverings.

  • Research Products

    (42 results)

All Other

All Publications (42 results)

  • [Publications] K.Kawarabayashi: "Subgraphs of graphs on surfaces with high representativity"J.Combin.Theory Ser.B. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kawarabayashi: "2-Connected 7-coverings of 3-connected graphs on surfaces"J.Graph Theory. 43. 26-36 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Enomoto: "Choice number of some complete multi-partite graphs"Discrete Math.. 244. 55-66 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kawarabayashi: "Path factors in cubic graphs"J.Graph Theory. 39. 188-193 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Ota: "Vertex-disjoint stars in graphs"Discuss.Math.Graph Theory. 21. 179-185 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Egawa: "Vertex-disjoint paths in graphs"Ars Combin.. 61. 23-31 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Kurosawa: "q-Linear functions and algebraic independence"Tokyo J.Math.. 25. 459-472 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Enomoto: "On the decomposition dimension of trees"Discrete Math.. 252. 219-215 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Enomoto: "Cyclic chromatic number of 3-connected plane graphs"SIAM J.Discrete Math.. 14. 121-137 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Ozawa: "Optimality and constructions of incomplete split-block designs"J.Statist.Plann.Inference. 106. 135-157 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Jimbo: "Discrete structure of group divisible designs with r=λ_1+1"Bull.Inst.Combin.Appl.. 32. 29-36 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] I.Ishii: "A new condition for a Heegaard splitting to be reducible"Japan.J.Math.(N.S.). 27. 327-352 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Nakamoto: "Diagonal flips in graphs on closed surfaces with specified face size distributions"Yokohama Math.J.. 49. 171-180 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] C.Cortes: "Diagonal flips in outer-triangulations on closed surfaces"Discrete Math.. 254. 63-74 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Kitakubo: "Re-embedding structures of 5-connected projective-planar graphs"Discrete Math.. 244. 211-221 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Nakamoto: "Generating triangulations on closed surfaces with minimum degree at least 4"Discrete Math.. 244. 345-349 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Ando: "On quadrangulations of closed surfaces covered by vertices of degree 3"Ars Combin.. 62. 121-127 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Negami: "Triangulations on closed surfaces covered by vertices of given degree"Graphs Combin.. 17. 529-537 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Negami: "Crossing numbers of graph embedding pairs on closed surfaces"J.Graph Theory. 36. 8-23 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Omori: "Singular systems of exponential functions"Math.Phys.Stud.. 23. 169-186 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Tamura: "On a fluctuation identity for multidimensional Levy processes"Tokyo J.Math.. 25. 363-380 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kawarabayashi, A.Nakamoto and K.Ota: "Subgraphs of graphs on surfaces with high representativity"J.Combin.Theory Ser.B. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Kawarabayashi, A.Nakamoto and K.Ota: "2-Connected 7-coverings of 3-connected graphs on surfaces"J.Graph Theory. 43. 26-36 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Enomoto, K.Ohba, K.Ota and J.Sakamoto: "Choice number of some complete multi-partite graphs"Discrete Math.. 244. 55-66 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Kawarabayashi, H.Matsuda, Y.Oda and K.Ota: "Path factors in cubic graphs"J.Graph Theory. 39. 188-193 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Ota: "Vertes-disjoint stars in graphs"Discuss.Math.Graph Theory. 21. 179-185 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Egawa and K.Ota: "Vertex-disjoint paths in graphs"Ars Combin.. 61. 23-31 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Kurosawa and I.Shiokawa: "q-Linear functions and algebraic independence"Tokyo J.Math.. 25. 459-472 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Enomoto and T.Nakamigawa: "On the decomposition dimension of trees"Discrete Math.. 252. 219-225 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Enomoto, M.Hornak and S.Jendrol: "Cyclic chromatic number of 3-connected plane graphs"SIAM J.Discrete Math.. 14. 121-137 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Ozawa, M.Jimbo, S.Kageyama and S.Mejza: "Optimality and constructions of incomplete split-block designs"J.Statist.Plann.Inference. 106. 135-157 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Jimbo and S.Kageyama: "Discrete structure of group divisible designs with r=λ+1"Bull.Inst.Combin.Appl.. 32. 29-36 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] I.Ishii: "A new condition for a Heegaard splitting to be reducible"Japan. J.Math. (N.S.). 27. 327-352 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.Nakamoto and S.Negami: "Diagonal flips in graphs on closed surfaces with specified face size distributions"Yokohama Math J.. 49. 171-180 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] C.Cortes, C.Grima, A.Marquez and A.Nakamoto: "Diagonal flips in outer-triangulations on closed surfaces"Discrete Math.. 254. 63-74 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Kitakubo and S.Negami: "Re-embedding structures of 5-connected projective planar graphs"Discrete Math.. 244. 211-221 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.Nakamoto and S.Negami: "Generating triangulations on closed surfaces with minimum degree at least 4"Discrete Math.. 244. 345-349 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Ando and A.Nakamoto: "On quadrangulations of closed surfaces covered by vertices of degree 3"Ars Combin.. 62. 121-127 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Negami and A.Nakamoto: "Triangulations on closed surfaces covered by vertices of given degree"Graphs Combin.. 17. 529-537 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Negami: "Crossing numbers of graph embedding pairs on closed surfaces"J.Graph Theory. 36. 8-23 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Omori, Y.Maeda, N.Miyazaki and A.Yoshioka: "Singular systems of exponential functions"Math.Phys.Stud.. 23. 169-186 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Tamura and H.Tanaka: "On a fluctuation identity for multidimensional Levy processes"Tokyo J.Math.. 25. 363-380 (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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