Research on Geometric and Combinatorial Structures

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Triangulations and Quadrangulations of a closed surface with sufficiently fine mesh will approximate the discrete structure of the surface. We first consider the problem to transform one triangulation (or quadrangulation) to another by a sequence of local deformations. As a result for triangulations, if these graphs have same number of vertices and the number is large enough, namely, these graphs are good approximation of the surfacs, then these can be transformed each other by a sequence of diagonal flips. On the other hand, for quadrangulations, there exists an discrete invariant "cycle parity" that is closely related to the structure of the mapping class group of the surface.In a graph embedded in a closed surface, the degree of a vertex is considered as a discrete curvature on that place. In a 3-connected graph on the sphere, we studied the minimum degree sum of a connected subgraph having prescribed order.We also consider the problem of embedding graphs in a book-type manifold. We gave good bounds for the pagenumber of complete bipartite graphs, where the pagenumber of a graph is the minimum number of pages in which one can embed the graph without crossing edges and the spine. For general graphs, we gave a lower bound for the number of crossing points of edges and the spine.Also, we investigate the following topics ; a presentation of mapping class groups in terms of Artin groups, Galois action on fundamental groups, study of difference sets using number theory, searching irreducible polynomials, isometric embedding of simplexes. and the problem to represent an arbitrary graph as an integral distance graph in the plane.