Every 3-connected essentially 10-connected line graph is Hamilton-connected

Thomassen conjectured that every 4-connected line graph is Hamiltonian. Lai et al. (in 2006) [5] considered whether the high essential connectivity of the 3-connected line graphs can guarantee the existence of the Hamiltonian cycle in graphs and they showed that every 3-connected, essentially 11-connected line graph is Hamiltonian. In this note, we show that every 3-connected, essentially 10-connected line graph is Hamiltonian-connected. The result strengthens the known result mentioned above.