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Hamilton-connectivity of line graphs with application to their detour index
Journal of Applied Mathematics and Computing ( IF 2.2 ) Pub Date : 2021-06-02 , DOI: 10.1007/s12190-021-01565-2
Yubin Zhong , Sakander Hayat , Asad Khan

A graph is called Hamilton-connected if there exists a Hamiltonian path between every pair of its vertices. Determining whether or not a graph is Hamilton-connected is an NP-complete problem. In this paper, we present two methods to show Hamilton-connectivity in graphs. The first method uses the vertex connectivity and Hamiltoniancity of graphs, and, the second is the definition-based constructive method which constructs Hamiltonian paths between every pair of vertices. By employing these proof techniques, we show that the line graphs of the generalized Petersen, anti-prism and wheel graphs are Hamilton-connected. Combining it with some existing results, it shows that some of these families of Hamilton-connected line graphs have their underlying graph families non-Hamilton-connected. This, in turn, shows that the underlying graph of a Hamilton-connected line graph is not necessarily Hamilton-connected. As side results, the detour index being also an NP-complete problem, has been calculated for the families of Hamilton-connected line graphs. Finally, by computer we generate all the Hamilton-connected graphs on \(\le 7\) vertices and all the Hamilton-laceable graphs on \(\le 10\) vertices. Our research contributes towards our proposed conjecture that almost all graphs are Hamilton-connected.



中文翻译:

线图的汉密尔顿连通性及其绕行指数的应用

如果在每对顶点之间都存在一条哈密顿路径,则一个图称为哈密顿连通图。确定一个图是否是 Hamilton-connected 是一个 NP-complete 问题。在本文中,我们提出了两种方法来显示图中的汉密尔顿连通性。第一种方法使用图的顶点连通性和哈密顿性,第二种方法是基于定义的构造方法,它在每对顶点之间构造哈密顿路径。通过使用这些证明技术,我们证明了广义彼得森、反棱镜和轮图的线图是汉密尔顿连通的。将其与现有的一些结果相结合,可以看出这些汉密尔顿连通线图族中的一些具有它们的底层图族非汉密尔顿连通。这反过来,表明哈密顿连通线图的底层图不一定是哈密顿连通的。作为附带结果,绕行指数也是一个 NP 完全问题,已为汉密尔顿连通线图的族计算。最后,通过计算机我们生成所有的 Hamilton-connected graph on\(\le 7\)顶点和\(\le 10\)顶点上的所有 Hamilton-laceable 图。我们的研究有助于我们提出的猜想,即几乎所有的图都是汉密尔顿连通的。

更新日期:2021-06-02
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