Combinatorial Theory

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We obtained high level results in graph theory.In particular, we proved the existence of k-facors, connected [2, k]-factors, long cycles or hamiltonian cycles assuming various kinds of connectivity (vertex-connectivity, edge-connectivity, toughness, binding number etc.). For example, we proved that with few exceptios the length of a logest cycle is at least the length of a logest path-1 if the sum of degrees of three independent vertices is at least the order of the graph. There is a strong connection between geometric properties of a surface and those of graphs enbedded in it. Geometric dual of a planar graph is well-defined, but it is not obvious what is the dual of graphs enbedded in other surfaces. We investigated duals of graphs enbedded in a projective plane. We also obtained properties of triangulations and quadrangulations of various surfaces, and decided the number of equivalence classes of such graphs under fundamental deformations. On algebraic combinatorial theory, we studied distance-regular graphs, association schemes, spin models, and Hadamard matrices. We clarified the relation of spin models and association schemes, and the properties of the character table of an association scheme. Also, we get various results on finite geometris, arrangements of lines or hyperplanes, computational geometries, and graph algorithms.