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 $t.$ We prove that every $k$-connected $[k+1,2]$-graph is hamiltonian-connected except $kK_1\vee G_{k},$ where $k\ge 2$ and $G_{k}$ is an arbitrary graph of order $k.$ This generalizes the Chvátal-Erdős theorem.