Accordion graphs: Hamiltonicity, matchings and isomorphism with quartic circulants

TL;DR

This paper introduces a new class of quartic graphs called accordion graphs, explores their Hamiltonian and matching properties, and characterizes when they are circulant graphs, extending prior work on cubic graphs.

Contribution

The paper defines accordion graphs, analyzes their PH-property, and characterizes their circulant nature, expanding understanding of Hamiltonian properties in quartic graphs.

Findings

A new class of quartic graphs called accordion graphs is introduced.

Certain accordion graphs are shown to have the PH-property, including some previously studied circulants.

Accordion graphs are circulant if and only if either n or k is odd, or both are even with k<4.

Abstract

Let $G$ be a graph of even order and let $K_{G}$ be the complete graph on the same vertex set of $G$. A pairing of a graph $G$ is a perfect matching of the graph $K_{G}$. A graph $G$ has the Pairing-Hamiltonian property (for short, the PH-property) if for each one of its pairings, there exists a perfect matching of $G$ such that the union of the two gives rise to a Hamiltonian cycle of $K_G$. In 2015, Alahmadi \emph{et al.} gave a complete characterisation of the cubic graphs having the PH-property. Most naturally, the next step is to characterise the quartic graphs that have the PH-property. In this work we propose a class of quartic graphs on two parameters, $n$ and $k$, which we call the class of accordion graphs $A[n,k]$. We show that an infinite family of quartic graphs (which are also circulant) that Alahmadi \emph{et al.} stated to have the PH-property are, in fact, members of this general class of accordion graphs. We also study the PH-property of this class of accordion graphs, at times considering the pairings of $G$ which are also perfect matchings of $G$. Furthermore, there is a close relationship between accordion graphs and the Cartesian product of two cycles. Motivated by a recent work by Bogdanowicz (2015), we give a complete characterisation of those accordion graphs that are circulant graphs. In fact, we show that $A[n,k]$ is not circulant if and only if both $n$ and $k$ are even, such that $k\geq 4$.