The hamiltonicity of essentially 9-connected line graphs

TL;DR

This paper proves that 3-connected, essentially 9-connected line graphs are Hamilton-connected, extending previous results and using quasigraphs and discharging techniques, with implications for claw-free graphs.

Contribution

It establishes a new connectivity condition ensuring Hamilton-connectedness in line graphs, strengthening prior bounds and extending to claw-free graphs.

Findings

3-connected, essentially 9-connected line graphs are Hamilton-connected

Method combines quasigraphs with discharging technique

Result applies to claw-free graphs

Abstract

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.