Minors of non-hamiltonian polyhedra and the Herschel family

TL;DR

This paper proves that all non-hamiltonian polyhedra contain the Herschel graph as a minor, establishing Herschel as the unique minimal non-hamiltonian polyhedron and characterizing certain minors.

Contribution

It provides a unified, shorter proof that non-hamiltonian polyhedra always contain the Herschel graph as a minor, and resolves a conjecture about minors with no K_{2,6} in such polyhedra.

Findings

Herschel graph is the unique minor-minimal non-hamiltonian polyhedron.

Every non-hamiltonian polyhedron contains the Herschel graph as a minor.

Characterization of non-hamiltonian polyhedra with no K_{2,6} minor.

Abstract

We show that every non-hamiltonian polyhedron contains the Herschel graph as a minor, implying that the Herschel graph is the unique minor-minimal non-hamiltonian polyhedron. Our approach unifies many previously known results on minors of non-hamiltonian polyhedra, while strengthening them with significantly shorter, non-computer-assisted proofs. As an application, we characterize non-hamiltonian polyhedra with no $K_{2,6}$ minor, resolving a conjecture of Ellingham, Marshall, Ozeki, Royle, and Tsuchiya.