Generalized honeycomb torus

Hamilton paths in Cayley graphs on generalized dihedral groups

“…A family of trivalent vertex-transitive graphs that have garnered attention over the last thirty-five years have been called brick products in [1,2,5], honeycomb tori in [8,9,12,13,14,15], honeycomb toroidal graphs in [3], and hexagonal toroidal embeddings in [6,11]. Altshuler [6] studied them when he was considering Hamilton cycles in graphs embedded in the torus.…”

Generalized honeycomb torus is Hamiltonian

“…For fundamental graph-theoretical terminology the reader is referred to [4]. Definition 1 [3]. Let n be a positive even integer, m be a positive integer, and d be a nonnegative integer that is less than n and is of the same parity as m. …”

“…1 for two examples of generalized honeycomb torus. It is known [3] that generalized honeycomb torus includes honeycomb torus, honeycomb rectangular torus, and honeycomb parallelogramic torus as special cases. For convenience, let us introduce the following notations.…”

“…Due to lower node degree and lower implementation cost than those of a standard torus of the same size, these architectures have incurred great research interest [1,2,5,6,[8][9][10][11]. Cho and Hsu [3] found that all these honeycomb torus networks can be characterized in a unified way, and thereby proposed a class of interconnection networks known as the generalized honeycomb torus.…”

“…Cho and Hsu [2] discovered a Hamiltonian cycle for faulty honeycomb rectangular torus. As for generalized honeycomb torus, Cho and Hsu [3], and Yang and Megson [12] proved the existence of Hamiltonian cycles for some special cases. Furthermore, Cho and Hsu [3] speculated that every generalized honeycomb torus is Hamiltonian.…”