无爪图及其扩展图的因子的研究

In this project, first we construct Z closure and N closure respectively, and prove that the two constructed closures can protect the existence of even factors of claw-free graphs and we improve the existed closures of claw-free graphs, which contain cycle closure, edge closure and *-closure, to make them also protect the existence of even factors of claw-free graphs. Then we mainly study the number and circumference of components, and the number of vertices of any maximum independent set in each component of even factors of claw-free graphs, and by the above closures and path factor construction, study the sufficient conditions, which make claw-free graphs contain some special path factors. Secondly, we prove that the neighborhood equivalence closure for general graphs can protect the existence of even factors of the generalizations of claw-free graphs (quasi-claw-free graphs and almost claw-free graphs), then we use the closure to study the number and circumference of components, and the number of vertices of any maximum independent set in each component of even factors of generalizations of claw-free graphs. Finally, we use the neighborhood equivalence closure and path factor construction to study the sufficient conditions, which make claw-free graphs contain some special path factors. At present, most of the research on claw-free graphs focuses on the properties of the special factors-the connected 2-factors, i.e., Hamiltonian properties. In this project, we mainly study the properties of general factors of claw-free graphs and its generalizations. Our project will enrich the research of claw-free graphs and its generalizations.

本项目分别构造Z闭包，N闭包，证明其能保证无爪图偶因子的存在性，并改进无爪图已有的圈闭包，边闭包，*-闭包使其同样保证无爪图偶因子的存在性;然后分别利用上述闭包研究无爪图的偶因子的分支数，周长，各分支所含任意最大独立集顶点数，并分别利用上述闭包及直接构造路因子的方法研究无爪图含有某些特殊路因子的充分条件；还证明对一般图均适用的邻域等价闭包能保证无爪图的扩展图（半无爪图，拟无爪图）的偶因子的存在性，并利用邻域等价闭包研究无爪图的扩展图的偶因子的分支数，周长，以及各因子分支含任意最大独立集顶点数；然后再分别利用邻域等价闭包，直接构造路因子的方法给出无爪图的扩展图含某些特殊路因子的充分条件。目前无爪图及其扩展图的研究结果大多是关于特殊的因子—连通的2-因子的性质，即Hamilton性质，本项目主要研究无爪图及其扩展图的较为一般的因子的性质，丰富了无爪图及其扩展图的研究理论。

本项目利用无爪图的Ryjacek闭包证明了一个连通的无爪图含有k叶-生成树当且仅当其Ryjacek闭包含有k叶-生成树，利用无爪图的闭包解决无爪图的某些因子的性质问题；证明了任意3-连通的几乎局部连通无爪图为hamilton连通图；证明了若一个连通的矩形连通的最小度数至少为5且不含有某两种特殊子图的无爪图是顶点泛圈图；分别给出了一个k-连通的阶数为n的几乎无爪图，半无爪图含有3叶-生成树的充分条件；给出了一个四双星树的最小和最大极图，并得到了四双星树的谱半径的上界。本项目的研究丰富了无爪图及其扩展图的研究理论。