Proof of a conjecture on hamiltonian-connected graphs

TL;DR

This paper proves that all 3-connected claw-free graphs with a domination number of at most 3 are hamiltonian-connected, confirming a conjecture and highlighting the structural properties that guarantee Hamiltonian connectivity.

Contribution

It establishes a sharp condition linking connectivity, claw-freeness, and domination number to Hamiltonian connectivity, confirming a conjecture from 2020.

Findings

Every 3-connected claw-free graph with domination number ≤ 3 is hamiltonian-connected.

The result is sharp, indicating the boundary conditions are optimal.

Supports the conjecture posed by Zheng et al. in 2020.

Abstract

We prove that every 3-connected claw-free graph with domination number at most 3 is hamiltonian-connected. The result is sharp and it is inspired by a conjecture posed by Zheng, Broersma, Wang and Zhang in 2020.