Perfect matchings, Hamiltonian cycles and edge-colourings in a class of cubic graphs

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