A graph $G$ has, for each of its perfect matches, a perfect match Hamiltonian property (PMH- properties). $G$. The study of graphs with PMH properties, initiated in the 1970s by Las Vergnas and H\”{a}ggkvist, combines three well-studied properties of graphs: matching, Hamiltonicity, and edge coloring. Study these notions of a cubic graph to characterize a cubic graph in which every exact match corresponds to one of the colors of the appropriate three-sided color scheme of the graph. ), that is, all 2-factors in the graph contain only even cycles. 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. It is to introduce an infinite family of graphs.Papillon graphs are coined, two parameters that determine the value of each parameter that these graphs have PMH properties or are simply E2F.Also, two parameters with different parameters We also show that the Papillon graph is not isomorphic.

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