Symmetries of the Honeycomb toroidal graphs

TL;DR

This paper determines the full automorphism groups of honeycomb toroidal graphs, a family of cubic graphs embedded on a torus, thereby solving two open problems and advancing understanding of their symmetries.

Contribution

It provides a complete characterization of the automorphism groups for all honeycomb toroidal graphs, addressing previously open research questions.

Findings

Full automorphism groups of honeycomb toroidal graphs are characterized.

Solved two open problems on graph symmetries.

Enhanced understanding of the symmetry structure of these graphs.

Abstract

{\em Honeycomb toroidal graphs} are a family of cubic graphs determined by a set of three parameters, that have been studied over the last three decades both by mathematicians and computer scientists. They can all be embedded on a torus and coincide with the cubic Cayley graphs of generalized dihedral groups with respect to a set of three reflections. In a recent survey paper B. Alspach gathered most known results on this intriguing family of graphs and suggested a number of research problems regarding them. In this paper we solve two of these problems by determining the full automorphism group of each honeycomb toroidal graph.