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

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.