玫瑰花窗图R_3k(1,3)的交叉数

The Crossing Number of Rose Windows Graph R_3k(1,3)

摘要: 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(1,3)的交叉数进行研究。首先根据好的画法得到R_3k(1,3)的交叉数上界；再将R_3k(1,3)的边集分成边不相交的3k组，利用反证法和数学归纳法，讨论所有可能情况，证得R_3k(1,3)的交叉数下界至少是2k，从而得到cr(R_3k(1,3))≥2k，k≥3。

Abstract: 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 proving it, the progress of domestic and foreign research on the field of crossing number of graphs has been slow. This paper focuses on the study of the crossing number of rose window graphs R_3k(1,3) . Firstly, we get the upper bound of the crossover number of R_3k(1,3) based on the well-drawn method; then divide the set of edges of R_3k(1,3) into the 3k groups whose edges do not intersect, and using the inverse method and the mathematical induction method, all possible cases are discussed, so that we can prove that the lower bound of the crossover number of R_3k(1,3) is at least 2k, and thus we can prove that cr(R_3k(1,3))≥2k , k≥3 .