GENERAL RESEARCH OF GRAPH THEORY

我们不仅学习了图论，而且还学习了它的应用和相关的数学分支。例如，我们研究了以下问题，并得到了以下结果。(1)图的匹配的自然推广：（1，f）-奇子图问题;（2）图在平面上的直线嵌入问题;（3）在三页书上嵌入图的最小交叉数的确定问题;（4）图有长圈的充分条件;（5）不含3-圈的有向图中4-圈的个数：（6）证明了对整数k>=3，每个最小度至少为2k且阶至少为g（k）的图都含有k个点不相交的等长圈;（7）给出了Klein瓶三角剖分具有汉密尔顿圈的一个充分条件，并研究了Klein瓶三角剖分在对角变换下的等价性;（8）图存在具有某种给定性质的圈的充分度条件;（9）图的带宽在矩阵理论中起着重要的作用。我们还学习了代数图论、组合数学、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的本原对称结合方案，并利用理论研究给出了一种有效的算法，可以检测出数字图像中包含的所有直线分量。

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.