AI 中文总结
本文针对k≥2，证明了2k-连通、1-坚韧且不含(P₃∪kP₁)子图的图必为哈密顿图，(2k+1)-连通、坚韧度大于1且不含(P₃∪kP₁)子图的图必为哈密顿连通图。
AI 中文摘要
若图G拥有哈密顿圈，则称G为哈密顿图；若G中任意两个不同顶点间都存在哈密顿路径，则称G为哈密顿连通图。非完全图的坚韧度是指，对任意割集S，|S|与G-S的连通分支数的最小比值。对给定图H，若图G不含H作为诱导子图，则称G为H-自由图。本文针对整数k≥2，证明了每一个2k-连通、1-坚韧且不含(P₃∪kP₁)子图的图都是哈密顿图，且每一个(2k+1)-连通、坚韧度大于1且不含(P₃∪kP₁)子图的图都是哈密顿连通图。
英文摘要
A graph $G$ is called Hamiltonian if it possesses a Hamiltonian cycle; and $G$ is called Hamiltonian-connected if it contains a Hamiltonian path between any two distinct vertices. The toughness of a non-complete graph is the minimum ratio of $|S|$ to the number of components of $G-S$ for any cutset $S$. For a given graph $H$, a graph $G$ is called $H$-free if $G$ does not contain $H$ as an induced subgraph. In this paper, for an integer $k\ge 2$, we prove that every $2k$-connected, $1$-tough and $(P_{3}\cup kP_{1})$-free graph is Hamiltonian and every $(2k+1)$-connected $(P_{3}\cup kP_{1})$-free graph with toughness greater than $1$ is Hamiltonian-connected.
