Prism‐hamiltonicity of triangulations

Abstract

Abstract:The prism over a graph G is the Cartesian product G K 2 of G with the complete graph K 2 . If the prism over G is hamiltonian, we say that G is prism-hamiltonian. We prove that triangulations of the plane, projective plane, torus, and Klein bottle are prism-hamiltonian. We additionally show that every 4-connected triangulation of a surface with sufficiently large representativity is prism-hamiltonian, and that every 3-connected planar bipartite graph is prism-hamiltonian.

Non‐rainbow colorings of 3‐, 4‐ and 5‐connected plane graphs

“…Hence, by Lemma 1, we have We assume that n ≡ 3 (mod 8) in the remainder of the proof. It is straightforward to verify that unless H is a single vertex or n ′ = 4, the estimates established in the previous paragraph yield w 4 …”

Hamilton cycles in prisms

“…Note that during the evaluation period of this article, Biebighauser and Ellingham [3] proved a stronger result that every plane triangulation is prism-hamiltonian. Therefore, we only sketch the proof of our result (as our proof method is different from that in [3]) and leave some of the details to the reader.…”

“…Note that during the evaluation period of this article, Biebighauser and Ellingham [3] proved a stronger result that every plane triangulation is prism-hamiltonian. Therefore, we only sketch the proof of our result (as our proof method is different from that in [3]) and leave some of the details to the reader. A kleetope is a plane graph obtained from a drawing of the complete graph K 4 by successive subdivisions of internal faces, that is, adding a new vertex to a face and joining it to all the three vertices on the boundary of the face.…”

“…In Section 3, we prove that all kleetopes, that is, 3-connected planar chordal graphs, are prism-hamiltonian. During the evaluation period of this article, Biebighauser and Ellingham [3] proved Conjecture A of Section 1 for planar triangulations and also extended their result to other surfaces: the projective plane, the torus, and the Klein bottle. In addition, they also proved that planar 3-connected bipartite graphs are prism-hamiltonian.…”

One‐way infinite 2‐walks in planar graphs

“…In this final section, we discuss spanning paths in prisms over infinite planar graphs, providing some partial results on the first part of Question 1.8. In [3], we showed that prisms over bipartite circuit graphs and near-triangulations are Hamiltonian. A near-triangulation is a finite plane graph where every face is a triangle, except for possibly the outer face, which is bounded by a cycle.…”