GENERAL RESEARCH OF GRAPH THEORY

我们不仅学习了图论，还学习了图论的应用和相关的数学分支。例如，我们研究了以下问题，得到了以下结果。(1)关于(1,f) -奇子图的问题，这是图的匹配的自然推广；(2)平面上给定点集上图的直线嵌入问题；(3)在三页书中嵌入图时，确定最小交叉数的问题；(4)图具有长周期的充分条件；(5)计算没有3环的有向图中4环的个数；(6)证明了对于整数k>=3，每一个最小度至少为2k且阶至少为g (k)的图都包含相同长度的顶点不相交的k个环；(7)得到了Klein瓶三角剖分具有Hamilton循环的充分条件，并研究了Klein瓶三角剖分在对角变换下的等价性；(8)图具有具有某种给定性质的环的充分次条件；(9)图的带宽在矩阵理论中占有重要的地位。我们确定树的带宽的上界。我们还学习了代数图论、组合学、离散几何、算法理论、数字图像的理论与算法、机器发现的数学理论。例如，我们研究了自旋模型，确定了m_1=3的原始对称关联方案；并在理论研究的基础上，提出了一种有效的检测数字图像中包含的每条线分量的算法。

We studied not only graph theory but also its applications and related branches of mathematics. For example, we researched the following problems and obtained the following results.(1) Problems on (1, f) -odd subgraphs, which are natural generalizaion of matchings of graphs ; (2) Problem on straight-line embedding of graphs onto a given set of points in the plane ; (3) Problem of determining the minimum crossing number when we embed a graph on a book in three pages ; (4) Sufficient conditions for a graph to have a long cycle ; (5) Counting the number of 4-cycles in a directed graph without 3-cycles ; (6) We prove that for an integer k>=3, every graph with minimum degree at least 2k and with order at least g (k) contains vertex disjoint k cycles of the same length ; (7) We obtain a sufficient condition for a triangulation of Klein bottle to have Hamilton cycle, and study the equivalence of triangulations of Klein bottle under diagonal tranformations ; (8) A sufficient degree condition for a graph to have a cycle possessing a certain given property ; (9) The bandwidth of a graph plays an important role in matrix theory. We determine the upper bound of the bandwidths of trees.We also studied algebraic graph theory, combinatorics, discrete geometry, algorithm theory, theory and algorithm for digital images, and mathematical theory of machine discovery. For example, we researched spin models, and detemine the primitive symmetric association schemes with m_1=3 ; and gave an effective algorithm for detecting every line component contained in a digital image by making use of thoretical studies.