On 3-polytopes with non-Hamiltonian prisms

Špacapan recently showed that there exist 3-polytopes with non-Hamiltonian prisms, disproving a conjecture of Rosenfeld and Barnette. By adapting Špacapan's approach we strengthen his result in several directions. We prove that there exists an infinite family of counterexamples to the Rosenfeld–Barnette conjecture, each member of which has maximum degree 37, is of girth 4, and contains no odd-length face with length less than (Formula presented.) for a given odd integer (Formula presented.). We also show that for any given 3-polytope (Formula presented.) there is a counterexample containing (Formula presented.) as an induced subgraph. This yields an infinite family of non-Hamiltonian 4-polytopes in which the proportion of quartic vertices tends to 1. However, Barnette's conjecture stating that every 4-polytope in which all vertices are quartic is Hamiltonian still stands. Finally, we prove that the Grünbaum–Walther shortness coefficient of the family of all prisms of 3-polytopes is at most 59/60.