On two graph isomorphism problems

TL;DR

This paper characterizes when 4-regular circulant graphs are isomorphic to accordion graphs, extending previous work on graph isomorphism conditions and providing criteria for graph isomorphism within this class.

Contribution

The authors provide a complete characterization of 4-regular circulant graphs that are isomorphic to accordion graphs and establish conditions for isomorphism between different accordion graphs.

Findings

Characterization of parameters for which an accordion graph is circulant.

Necessary and sufficient conditions for isomorphism between accordion graphs.

Extension of Bogdanowicz's work to a broader class of graphs.

Abstract

In 2015, Bogdanowicz gave a necessary and sufficient condition for a 4-regular circulant graph to be isomorphic to the Cartesian product of two cycles. Accordion graphs, denoted by $A[n,k]$, are 4-regular graphs on two parameters $n$ and $k$ which were recently introduced by the authors and studied with regards to Hamiltonicity and matchings. These graphs can be obtained by a slight modification in some of the edges of the Cartesian product of two cycles. Motivated by the work of Bogdanowicz, the authors also determined for which values of $n$ and $k$ the accordion graph $A[n,k]$ is circulant. In this work we investigate what parameters a 4-regular circulant graph must have in order to be isomorphic to an accordion graph, thus providing a complete characterisation similar to that given by Bogdanowicz. We also give a necessary and sufficient condition for two accordion graphs with distinct parameters to be isomorphic.