On Thomassen's conjecture and 2-factors of claw-free graphs

首先，利用最大独立数给出了低连通线图的2因子分量数的上界。得到如下结果：一个3连通的无爪图有2个因子，其中分量的个数不超过2a/5，其中a为最大独立数。接下来我们考虑了三次图，它是图论中对Thomassen猜想和许多重要问题和猜想非常重要的图族。我们证明了如果邦迪关于三次图的猜想成立，那么托马森的猜想对最小次数至少为5的情况成立。

First, we studied the upper bounds of the number of components of 2-factors of low-connected line graphs, which are given by using the maximum independence number. And we got the following result: a 3-connected claw-free graph has a 2-factor in which the number of components is at most 2a/5, where a is the maximum independence number. Next we considered cubic graphs, which is very important family of graphs for Thomassen's conjecture and many important problems and conjectures in Graph theory. And we showed that if Bondy's conjecture on cubic graphs holds, then Thomassen's conjecture holds for the case of the minimum degree at least five.