  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 $k\geq 4$.
