  A graph $G$ has the Perfect-Matching-Hamiltonian property (PMH-property) if
for each one of its perfect matchings, there is another perfect matching of $G$
such that the union of the two perfect matchings yields a Hamiltonian cycle of
$G$. The study of graphs that have the PMH-property, initiated in the 1970s by
Las Vergnas and H\"{a}ggkvist, combines three well-studied properties of
graphs, namely matchings, Hamiltonicity and edge-colourings. In this work, we
study these concepts for cubic graphs in an attempt to characterise those cubic
graphs for which every perfect matching corresponds to one of the colours of a
proper 3-edge-colouring of the graph. We discuss that this is equivalent to
saying that such graphs are even-2-factorable (E2F), that is, all 2-factors of
the graph contain only even cycles. The case for bipartite cubic graphs is
trivial, since if $G$ is bipartite then it is E2F. Thus, we restrict our
attention to non-bipartite cubic graphs. A sufficient, but not necessary,
condition for a cubic graph to be E2F is that it has the PMH-property. The aim
of this work is to introduce an infinite family of E2F non-bipartite cubic
graphs on two parameters, which we coin papillon graphs, and determine the
values of the respective parameters for which these graphs have the
PMH-property or are just E2F. We also show that no two papillon graphs with
different parameters are isomorphic.
