  Yang et al. proved that every 3-connected, essentially 11-connected line
graph is Hamilton-connected. This was extended by Li and Yang to 3-connected,
essentially 10-connected graphs. Strengthening their result further, we prove
that 3-connected, essentially 9-connected line graphs are Hamilton-connected.
We use a method based on quasigraphs in combination with the discharging
technique. The result extends to claw-free graphs.
