On perfect 2-matching uniform graphs

Let G be a graph. For a set H of connected graphs, an H-factor of graph G is a spanning subgraph H of G such that every component of H is isomorphic to a member of H. Denote H=₂\∪ ᵢ|i≥ 3\. We call H-factor a perfect 2-matching of G, that is, a perfect 2-matching is a spanning subgraph of G such that each component of G is either an edge or a cycle. In this paper, we define the new concept of perfect $2$-matching uniform graph, namely, a graph G is called a perfect $2$-matching uniform graph if for arbitrary two distinct edges e₁ and e₂ of G, G contains a perfect $2$-matching containing e₁ and avoiding e₂. In addition, we study the relationship between some graphic parameters and the existence of perfect $2$-matching uniform graphs. The results obtained in this paper are sharp in some sense.