On saturated non-covered graphs

Abstract

A connected simple graph $G$ of order $2n$ with edge set $E\left(G\right)$ is a matching covered graph if it has at least one edge and each edge of $G$ is contained in some perfect matching of $G$. Define a graph $G$ to be saturated non-covered if it is not matching covered, but $G+e$ for any $e\in E\left(G^c\right)$, the complement of $G$, is a matching covered graph. In this paper, we obtain a characterization of saturated non-covered graphs. Furthermore, we prove that $G$ of order $2n\ge 4$ is matching covered if one of the following statements holds: (1) the number of spanning trees of $G$ is either more than 324 for $n=3$ or more than $4n{(2n-1)}^{2n-4}$ for $n\ne 3$; (2) the Wiener index of $G$ is less than $2n^2+n-3$; (3) the Kirchhoff index of $G$ is less than $3n-2$.