The Minimum Number of Edges in $(p+1)K_2$-Saturated Graphs

Given a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-saturated if it is $\mathcal{F}$-free but the addition of any missing edge creates a copy of some $F\in \mathcal{F}$. The study of the minimum number of edges in $\mathcal{F}$-saturated graphs is a central topic in extremal graph theory. Let $(p+1)K_2$ denote a matching of size $p+1$. Determining the minimum number of edges in a $(p+1)K_2$-saturated graph is a fundamental question in this area, explicitly posed as Problem 9 in the survey by Faudree et al. (2011). In this paper, we refine the structural analysis of $(p+1)K_2$-saturated graphs and derive an explicit formula for the number of edges in terms of a single integer parameter. By minimizing this formula we determine $\mathrm{s}\mathrm{a}\mathrm{t}(n,(p+1)K_2)$ for all $n>2p$, thereby resolving Problem 9 in full generality and extending earlier results of Kászonyi–Tuza (1986) and Zhang–Lu–Yu (2024). Moreover, by maximizing the same formula we recover the classical Erdős–Gallai (1959) upper bound on the number of edges in such graphs.