The Minimum Number of Edges in $(p+1)K_2$-Saturated Graphs
Xiaoteng Zhou, Kazuya Haraguchi, Hanchun Yuan
TL;DR
This work determines the saturation number sat$(n,(p+1)K_2)$ for all integers $n>2p$, resolving a long-standing open problem and unifying prior results. The authors exploit a structural decomposition: a $(p+1)K_2$-saturated graph $G$ splits into a universal-vertex set $Z$ and a remainder $G-Z$ that is a disjoint union of odd cliques; with $k=n-2p$ and $z=|Z|$, they derive an edge-count expression that reduces to an optimization over $z$ and the sizes of the odd-clique components. They define a main term $g(z)$ and a remainder $R(z)$ so that $D(z)=g(z)+R(z)$ governs the minimum via $ ext{sat}(n,(p+1)K_2)= rac{D(z^*)}{2}$, where for $n>2k^2-2k$ the minimizer $z^*$ lies near $z^67=(\,\sqrt{8n+8k+1}-4k+1\,)/4$ (specifically $ loor{z^67}$ or $ ound{z^67}$), and for $n\le 2k^2-2k$ we have $z^*=0$. The results recover Erdős–Gallai bounds as a maximizing case and connect to KT (1986) and Zhang–Lu–Yu (2024) in the appropriate regimes, while providing a complete solution and enumerations for small cases ($n\le18$). The methodological framework opens avenues for extending saturation-number analysis to $K_r$-saturated graphs with $r\ge3$.
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 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.