On \(s\)-Hamiltonian-connected line graphs (Q941364)

A graph \(G\) is Hamiltonian-connected if any two of its vertices are connected by a Hamiltonian path; and is \(s\)-Hamiltonian-connected if the deletion of any vertex subset with at most s vertices results in a Hamiltonian-connected graph. Thomassen conjectured that every 4-connected line graph is Hamiltonian. In this paper the authors prove the following related result. Theorem. The line graph of a \((t+4)\)-edge-connected graph is \((t+2)\)-Hamiltonian-connected if and only if it is \((t+5)\)-connected, and for \(s\geq 2\) every \((s+5)\)-connected line graph is \(s\)-Hamiltonian-connected.