Common matching number of a graph

The cardinality of the largest matching in a graph $G$, denoted by ${\alpha }^{\prime }\left(G\right)$, is referred to as the upper matching number of $G$. The lower matching number $i^{\prime }\left(G\right)$ is defined as the cardinality of the smallest maximal matching in $G$. We introduce the concept of the common matching number of a graph $G$, denoted by ${\alpha }_c^{\prime }\left(G\right)$, which is the largest integer $k$ such that every edge in $G$ belongs to a matching that contains at least $k$ edges. In this paper, we explore the relationships between the parameters $i^{\prime }\left(G\right)$, ${\alpha }_c^{\prime }\left(G\right)$, and ${\alpha }^{\prime }\left(G\right)$. In particular, we demonstrate that the difference between ${\alpha }_c^{\prime }\left(G\right)$ and $i^{\prime }\left(G\right)$ can be arbitrarily large, while the difference between ${\alpha }^{\prime }\left(G\right)$ and ${\alpha }_c^{\prime }\left(G\right)$ can at most be one. Additionally, we characterize the trees $T$ for which $i^{\prime }\left(T\right)={\alpha }_c^{\prime }\left(T\right)$, as well as the trees $T$ for which ${\alpha }_c^{\prime }\left(T\right)={\alpha }^{\prime }\left(T\right)$.