Minimizing the number of matchings of fixed size in a $K_s$-saturated graph

TL;DR

This paper determines the minimal number of fixed-size matchings in large $K_s$-saturated graphs, identifying extremal structures and proving their uniqueness for certain parameters.

Contribution

It establishes the extremal $K_s$-saturated graphs minimizing matchings of size $k$, including uniqueness results for specific cases.

Findings

$S_{n,s-2}$ minimizes the number of $M_k$ in large $K_s$-saturated graphs for $k eq s-2$.

$S_{n,1}$ is the unique extremal graph for $k=2$, $s=3$.

The extremal graphs are characterized and proven to be unique under given conditions.

Abstract

For a fixed graph $F$, a graph $G$ is said to be $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$ but does contain $F$ after the addition of any new edge. Let $M_k$ be a matching consisting of $k$ edges and $S_{n,k}$ be the join graph of a complete graph $K_k$ and an empty graph $\overline{K_{n-k}}$. In this paper, we prove that for $s \geq3$ and $k\geq 2$, $S_{n,s-2}$ contains the minimum number of $M_k$ among all $n$-vertex $K_s$-saturated graphs for sufficiently large $n$, and when $k \leq s-2$, it is the unique extremal graph. In addition, we also show that $S_{n,1}$ is the unique extremal graph when $k=2$ and $s=3$.