Sufficient conditions for $(K_2 \cup kK_1)$-free graphs to be Hamilton-connected

The toughness of a non-complete graph $G$, denoted $\tau(G)$, is defined as $$\tau(G) = \min\left\{ \frac{|S|}{\omega(G-S)} : S \subseteq V(G),\ \omega(G-S) \geq 2 \right\},$$ where $\omega(G-S)$ is the number of components of $G - S$. For a complete graph $G$, we define $\tau(G) = \infty$. A graph $G$ is $t$-tough if $\tau(G) \geq t$. 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 $\tau(G) > 1$ 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 $\tau(G) > 1$ and minimum degree $\delta(G) \geq 2k$ is Hamilton-connected. Moreover, by imposing restrictions to the independence number $\alpha(G)$, we prove that every $k$-connected $(K_2 \cup kK_1)$-free graph $G$ of order $n$ with $2k+1 \leq \alpha(G) < \frac{n}{2}$ and $\delta(G) \geq 2k$ is Hamilton-connected, and that the bounds on $\alpha(G)$ are sharp.