On Hypohamiltonian Snarks and a Theorem of Fiorini

TL;DR

This paper corrects and generalizes Fiorini's 1983 theorem on hypohamiltonian snarks, establishing the existence of such graphs for all relevant orders and verifying related conjectures up to 36 vertices.

Contribution

It rectifies Fiorini's theorem, extends Steffen's results to all possible orders, and confirms a conjecture on hypohamiltonian snarks up to 36 vertices.

Findings

Hypohamiltonian snarks exist for all orders where they are possible.

The corrected theorem broadens the understanding of hypohamiltonian snarks.

Verification of Steffen's conjecture up to 36 vertices.

Abstract

We discuss an omission in the statement and proof of Fiorini's 1983 theorem on hypohamiltonian snarks and present a version of this theorem which is more general in several ways. Using Fiorini's erroneous result, Steffen showed that hypohamiltonian snarks exist for some $n \ge 10$ and each even $n \ge 92$. We rectify Steffen's proof by providing a correct demonstration of a technical lemma on flower snarks, which might be of separate interest. We then strengthen Steffen's theorem to the strongest possible form by determining all orders for which hypohamiltonian snarks exists. This also strengthens a result of M\'{a}\v{c}ajov\'{a} and \v{S}koviera. Finally, we verify a conjecture of Steffen on hypohamiltonian snarks up to 36 vertices.