Putting Tutte's counterexample to Tait's conjecture in perspective to hamiltonicity and non-hamiltonicity in certain planar cubic graphs

TL;DR

The paper addresses Tait's conjecture by situating Tutte's and Holton–McKay's counterexamples within an infinite family generated via TF-inflation on $n$-prisms $C_n\square K_2$. It develops a parity-based framework: inflated graphs are Hamiltonian for even $n$ and non-Hamiltonian for odd $n$, while also providing a precise Hamiltonian-cycle counting formula $h(q,q_1)=2^{q_1}4^{q-q_1}+4^{q_1}2^{q-q_1}$ after $q$ TF-inflations. By evaluating the case $C_3\square K_2$ with symmetric TF-inflations and contracting the top triangle, Tutte's counterexample is recovered, and the work shows these classical counterexamples appear as minimal elements in their respective inflation families. The results illuminate how TF-inflation shapes Hamiltonicity, producing both Hamiltonian and non-Hamiltonian planar cubic graphs and offering a systemic perspective on the origins of these longstanding counterexamples.

Abstract

Using the graphs of prisms and Tutte Fragments, we construct an infinite family of hamiltonian and non-hamiltonian graphs in which Tutte's counterexample to Tait's conjecture appears in a certain sense as a minimal element. We observe that generalizations of the minimum-cardinality counterexamples of Holton and McKay to Tait's conjecture are as well contained in this family.

Figures (3)

• Figure 1: Celebrated Tutte fragment (TF) found in $1946$. No hamiltonian path of the TF with endpoints $a$ and $b$ exists. However, certain hamiltonian paths with endpoint-pair $(a,c)$ as well as $(b,c)$ exist.
• Figure 2: Three pillars of an $n$-prism with single- and both-sided TF-inflations: A single TF-inflation at the pillar's top (respectively bottom) vertex is shown for the left (respectively middle) pillar. In case of the right pillar, both vertices have been subjected to a TF-inflation.
• Figure 3: Each hamiltonian cycle of $C_{n}\,\square\,K_2$ either covers exactly two consecutive or all pillars. Pillars that are uncovered in a given hamiltonian cycle are labeled $u_i$, $u'_i$.

Theorems & Definitions (11)

• Definition 1: TF-inflation
• Lemma 1: tutte1946hamiltonian Hamiltonian paths of the TF
• Lemma 2
• proof
• Definition 2
• Lemma 3
• proof
• Remark 1
• Theorem 1
• proof