Polyhedra of Small Order and Their Hamiltonian Properties

Abstract

We describe the results of an enumeration of several classes of polyhedra. The enumerated classes include polyhedra with up 13 vertices, simplicial polyhedra with up to 16 vertices, 4-connected polyhedra with up to 15 vertices, non-Hamiltonian polyhedra with up to 15 vertices, bipartite polyhedra with up to 24 vertices, and bipartite trivalent polyhedra with up to 44 vertices. The results of the enumeration were used to systematically search for certain smallest non-Hamiltonian polyhedral graphs. In particular, the smallest non-Hamiltonian planar graphs satisfying certain toughness-like properties are presented here, as are the smallest non-Hamiltonian, 3-connected, Delaunay tessellations and triangulations. Improved upper and lower bounds on the size of the smallest non-Hamiltonian, inscribable polyhedra are also given.

Construction of planar triangulations with minimum degree 5

“…Classes of polyhedra were among the first graph classes for which construction methods were published (see [7] and [11]) and also among the first classes for which a computer was used for their enumeration (see [9]). Today, the most extensive tables for various classes of polyhedra are given by Dillencourt in [6].…”

Hamiltonicity of planar graphs with a forbidden minor

“…It was known to C. N. Reynolds (in dual form) in 1931, as reported by Whitney [24, Figure ]. Again, the proof that this is smallest relies on Barnette and Jucovič and Dillencourt . This triangulation was also presented much later by Goldner and Harary , so it is sometimes called the Goldner‐Harary graph.…”

“…The smallest ‐connected planar graph that is not Hamiltonian is the so‐called Herschel graph, with vertices and edges. It was known to Coxeter in 1948 [6, p. 8], but a proof that it is smallest relies on later work by Barnette and Jucovič and Dillencourt . If we restrict to triangulations, the smallest ‐ or ‐connected planar triangulation that is not Hamiltonian is a triangulation obtained by adding nine edges to the Herschel graph.…”

Longer cycles in essentially 4-connected planar graphs

“…Let G be an essentially 4-connected plane graph. It is well-known that every 3connected plane graph on at most 10 vertices is Hamiltonian [1]; thus, for 4 ≤ n ≤ 10, this implies circ(G) = n ≥ 5 8 (n + 2). Since these graphs contain in particular the essentially 4-connected plane graphs on at most 10 vertices, we assume n ≥ 11 from now on.…”