On perfect 2-matching uniform graphs

Let $G$ be a graph. For a set $\mathcal{H}$ of connected graphs, an $\mathcal{H}$-factor of graph $G$ is a spanning subgraph $H$ of $G$ such that every component of $H$ is isomorphic to a member of $\mathcal{H}$. Denote $\mathcal{H}=\{P_2\}\cup \{C_i|i\ge 3\}$. We call $\mathcal{H}$-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 $e_1$ and $e_2$ of $G$, $G$ contains a perfect $2$-matching containing $e_1$ and avoiding $e_2$. 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.