Research on Combinatorial Geometry

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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.