玫瑰花窗图R(3k, 3, 2)的交叉数

The Crossing Number of Rose Windows Graph R(3k, 3, 2)

摘要: 1738年，瑞典数学家欧拉解决了哥尼斯堡七桥问题，图论由此诞生。图的交叉数是图论中一个重要的部分，近百年来，国内外很多学者都对图的交叉数这一问题进行研究，但由于证明难度较大，国内外关于图的交叉数领域的研究进展缓慢。本文主要对玫瑰花窗图R(3k, 3, 2)的交叉数进行研究。首先根据好的画法得到R(3k, 3, 2)的交叉数上界，再利用反证法和数学归纳法，将R(3k, 3, 2)的边集分成边不相交的3k组，讨论所有可能情况，证得R(3k, 3, 2)的交叉数下界至少是3k，从而证得cr(R(3k, 3, 2)) = 3k, k ≥ 4。

Abstract: In 1738, the Swedish mathematician Euler solved the problem of seven bridges in Königsberg, and graph theory was born. Crossing number of graphs is an important part of graph theory, and many scholars at home and abroad have studied the problem of crossing number of graphs in the past hundred years, but due to the difficulty of the proof, the research progress in the field of crossing number of graphs at home and abroad has been slow. In this paper, we mainly study the crossing number of rose window graph R(3k, 3, 2). Firstly, we get the upper bound of the crossover number of R(3k, 3, 2) based on the well-drawn method, and then we use the inverse method and the mathematical induction method to divide the set of edges of R(3k, 3, 2) into the 3k groups whose edges do not intersect, and discuss all the possible cases, so that we can prove that the lower bound of the crossover number of R(3k, 3, 2) is at least 3k, and thus we can prove that cr(R(3k, 3, 2)) = 3k, k ≥ 4.