A generalization of the Chvátal-Erdős theorem

A well-known result of Chvátal and Erdős from 1972 states that a graph with connectivity not less than its independence number plus one is hamiltonian-connected. A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$ of order $s$ has size at least $t\mathrm{.}$ We prove that every $k$-connected $[k+1,2]$-graph is hamiltonian-connected except $k{K}_1\vee G_k,$ where $k\ge 2$ and $G_k$ is an arbitrary graph of order $k\mathrm{.}$ This generalizes the Chvátal-Erdős theorem.