On 3-Polytopes With Non-Hamiltonian Prisms

Spacapan recently showed that there exist 3-polytopes with non-Hamiltonian prisms, disproving a conjecture of Rosenfeld and Barnette. By adapting Spacapan'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 k for a given odd integer k. We also show that for any given 3-polytope H there is a counterexample containing H 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 Grunbaum-Walther shortness coefficient of the family of all prisms of 3-polytopes is at most 59/60.