Combinatorial Theory

我们在图论中得到了高水平的结果，特别是在各种连通性（点连通性、边连通性、韧性、结合数等）的假设下，证明了k-因子、连通[2，k]-因子、长圈或Hamilton圈的存在性。例如，我们证明了：如果图的三个独立顶点的度之和至少是图的阶，则在少数例外情况下，logest圈的长度至少是logest路-1的长度。曲面的几何性质与嵌入其中的图的几何性质之间有很强的联系，平面图的几何对偶是有明确定义的，但嵌入其它曲面的图的几何对偶是什么并不明显。研究了嵌入在射影平面上的图的连通性。我们还得到了各种曲面的三角剖分和四边形剖分的性质，并确定了这类图在基本变形下的等价类数。在代数组合理论方面，我们研究了距离正则图、结合方案、自旋模型和阿达玛矩阵。阐明了自旋模型与缔合方案的关系，以及缔合方案特征标表的性质。此外，我们得到了各种结果有限geometris，安排线或超平面，计算几何和图形算法。

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.