| Project/Area Number |
11640135
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Keio University |
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Principal Investigator |
OTA Katsuhiro Keio University, Department of Mathematics, Associate Professor, 理工学部, 助教授 (40213722)
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| Co-Investigator(Kenkyū-buntansha) |
MAEDA Yoshiaki Keio University, Department of Mathematics, Professor, 理工学部, 教授 (40101076)
ENOMOTO Hikoe Keio University, Department of Mathematics, Professor, 理工学部, 教授 (00011669)
SHIOKAWA Iekata Keio University, Department of Mathematics, Professor, 理工学部, 教授 (00015835)
NAKAMOTO Atsuhiro Osaka Kyoiku University, Department of Mathematics, Instructor, 教育学部, 助手 (20314445)
JIMBO Masakazu Keio University, Department of Mathematics, Professor, 理工学部, 教授 (50103049)
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| Project Period (FY) |
1999 – 2000
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| Keywords | Topological graph theory / Triangulation / Quadrangulation / Chromatic number / Representativity / Hamiltonicity |
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| Research Abstract |
Triangulations and Quadrangulations of a closed surface are most typical combinatorial objects related with geometic ones. We first consider the problem to transform one triangulation (or quadrangulation) to another by a sequence of local deformations. Continued to the preceding research, we have obtained some results for degree constrained cases and for outer-triangulations, a generalization of outer planar graphs. We have also obtained an interesting result on planar triangulations which can be embedded as a quadrangulation of another surface. In particular, it depends on the orientability of the surface, and certain combinatorial structure of the triangulation plays a key role.
Coloring of quadrangulation has a similar phenomenon that appears in only nonorientable surfaces. If the quadrangulation has a cycle cutting the surface into orientable one, then the chromatic number is at least 4. In particular, the chromatic number of a quadrangulation of torus and Klein bottle is determined by a topological and algebraic invariant of the graph.
High representativity of a 3-connected graph G on a surface enable us to cut open G into a suitable plane graph. We have established a very useful tool describing such a cutting. Using this tool, we have proved several theorems that show 3-connected graphs on a surface are close to be hamiltonian in a sense ; concerning spanning tree with maximum degree at most four, spanning 2-connected subgraph with maximum degree at most eight, and light connected subgraphs with given size.
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