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

TL;DR

This paper determines the minimum number of edges in graphs that are saturated with respect to a matching of size p+1, solving a longstanding problem and extending classical bounds in extremal graph theory.

Contribution

The authors derive an explicit formula for the minimum edges in $(p+1)K_2$-saturated graphs and fully solve Problem 9, extending previous results.

Findings

Explicit formula for the number of edges in $(p+1)K_2$-saturated graphs.

Complete solution to Problem 9 for all $n > 2p$.

Recovery of Erd ext{"o}s--Gallai upper bound via maximization.

Abstract

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 $\mathrm{sat}(n,(p+1)K_2)$ for all $n>2p$, thereby resolving Problem 9 in full generality and extending earlier results of K\'aszonyi--Tuza (1986) and Zhang--Lu--Yu (2024). Moreover, by maximizing the same formula we recover the classical Erd\H{o}s--Gallai (1959) upper bound on the number of edges in such graphs.