Hamiltonian-connectedness of triangulations with few separating triangles

TL;DR

This paper proves that certain 3-connected triangulations with limited separating triangles are hamiltonian-connected, and explores the bounds of this property through theoretical and computational results.

Contribution

It establishes conditions under which 3-connected triangulations are hamiltonian-connected and identifies the limits of this property with counterexamples and computational evidence.

Findings

Triangulations with at most one separating triangle are hamiltonian-connected.

There exist 3-connected triangulations with s ≥ 4 separating triangles that are not hamiltonian-connected.

Small 3-connected triangulations with up to 3 separating triangles are all hamiltonian-connected.

Abstract

We prove that 3-connected triangulations with at most one separating triangle are hamiltonian-connected. In order to show bounds on the strongest form of this theorem, we proved that for any $s\geq4$ there are 3-connected triangulation with $s$ separating triangles that are not hamiltonian-connected. We also present computational results which show that all `small' 3-connected triangulations with at most 3 separating triangles are hamiltonian-connected.