Minors of non-hamiltonian polyhedra and the Herschel family

TL;DR

This work establishes that the Herschel graph $\mathfrak{H}$ is the unique minor-minimal non-hamiltonian polyhedron by proving that every non-hamiltonian polyhedral graph contains $\mathfrak{H}$ as a minor. The authors provide a streamlined, non-computer-assisted proof built on planar embeddings and sector attachments, unifying prior minor-structure results for non-hamiltonian polyhedra. They further classify non-hamiltonian polyhedral graphs that avoid the $K_{2,6}$ minor, showing they must contain a spanning subgraph from the Herschel family, and they resolve a conjecture by Ellingham et al. by characterizing such graphs as $G_n^\bullet$ or $G_n^\circ$ for large $n$, with a precise count of cases. The results clarify the role of Herschel-type structures in planar, 3-connected graphs and provide a tight minor-based dichotomy for Hamiltonicity in polyhedra, with implications for minor theory and graph structure in planar settings.

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.

Figures (12)

• Figure 1: The Herschel graph $\mathfrak{H}$.
• Figure 2: The Herschel family. In the drawing of $\mathfrak{H}_n^\bullet$ (respectively, $\mathfrak{H}_n^\circ$), the rightmost vertical path degenerates into a single vertex when $n = 11$ (respectively, $n = 13$).
• Figure 3: Illustrations for the cases in the proof of Lemma \ref{['lem']} demonstrating a minor of the Herschel graph in $G$.
• Figure 4: A minor of the Herschel graph in $G$ when $|M| = 2$.
• Figure 5: A minor of the Herschel graph in $G$ when $|M| = 4$.

Theorems & Definitions (39)

• Theorem 1.2: Ellingham, Marshall, Ozeki, and Tsuchiya Ellingham2019
• Conjecture 1.3: Ellingham, Marshall, Ozeki, Royle, and Tsuchiya O'Connell2018
• Theorem 1.4: Ellingham, Gaslowitz, O'Connell, and Royle O'Connell2018
• Conjecture 1.5: Ding and Marshall Ding2018
• Theorem 1.6: Ding and Marshall Ding2018
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1: Ding2018
• Lemma 2.2
• proof