Every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected

Abstract

Given a graph $H$, a graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. Shi and Shan conjectured that every $1$-tough $2k$-connected $(P_2 \cup kP_1)$-free graph is hamiltonian for $k \geq 4$. This conjecture has been independently confirmed by Xu, Li, and Zhou, as well as by Ota and Sanka. Inspired by this, we prove that every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected.