关于不含\((K_2 \cup kK_1)\)的图为哈密顿连通图的充分条件
Sufficient conditions for $(K_2 \cup kK_1)$-free graphs to be Hamilton-connected
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中文总结 AI 辅助
研究不含\((K_2 \cup kK_1)\)的图为哈密顿连通图的充分条件。通过加强连通度、最小度等条件，以及对独立数施加限制，证明了不同条件下此类图是哈密顿连通的，且部分条件界限是紧的。
中文摘要 AI 辅助
非完全图\(G\)的坚韧度\(\tau(G)\)定义为\(\tau(G) = \min\left\{ \frac{|S|}{\omega(G - S)}: S \subseteq V(G),\ \omega(G - S) \geq 2 \right\}\)，其中\(\omega(G - S)\)是\(G - S\)的连通分支数，完全图\(G\)时\(\tau(G) = \infty\)。若\(\tau(G) \geq t\)，则图\(G\)是\(t -\)坚韧的。若图\(G\)不含与\(K_2 \cup kK_1\)同构的导出子图，则称\(G\)是\((K_2 \cup kK_1)\) - 自由的。Liu表明每个\(2k -\)连通、\(\tau(G) > 1\)的\((K_2 \cup kK_1)\) - 自由图\(G\)是哈密顿连通的。本文证明了每个\((k + 1)\) - 连通、\(\tau(G) > 1\)且最小度\(\delta(G) \geq 2k\)的\((K_2 \cup kK_1)\) - 自由图\(G\)是哈密顿连通的。还通过对独立数\(\alpha(G)\)施加限制，证明了每个阶为\(n\)、\(2k + 1 \leq \alpha(G) < \frac{n}{2}\)且\(\delta(G) \geq 2k\)的\(k -\)连通\((K_2 \cup kK_1)\) - 自由图\(G\)是哈密顿连通的，且\(\alpha(G)\)的界是紧的。
英文摘要
The toughness of a non-complete graph $G$, denoted $τ(G)$, is defined as \[ τ(G) = \min\left\{ \frac{|S|}{ω(G-S)} : S \subseteq V(G),\ ω(G-S) \geq 2 \right\}, \] where $ω(G-S)$ is the number of components of $G - S$. For a complete graph $G$, we define $τ(G) = \infty$. A graph $G$ is $t$-tough if $τ(G) \geq t$. For a positive integer $k$, a graph $G$ is $(K_2 \cup kK_1)$-free if it contains no induced subgraph isomorphic to $K_2 \cup kK_1$. Recently, Liu \cite{liu} showed that every $2k$-connected $(K_2 \cup kK_1)$-free graph $G$ with $τ(G) > 1$ is Hamilton-connected. In this paper, we strengthen this result by proving that every $(k+1)$-connected $(K_2 \cup kK_1)$-free graph $G$ with $τ(G) > 1$ and minimum degree $δ(G) \geq 2k$ is Hamilton-connected. Moreover, by imposing restrictions to the independence number $α(G)$, we prove that every $k$-connected $(K_2 \cup kK_1)$-free graph $G$ of order $n$ with $2k+1 \leq α(G) < \frac{n}{2}$ and $δ(G) \geq 2k$ is Hamilton-connected, and that the bounds on $α(G)$ are sharp.
