An Erd\H{o}s-Ko-Rado type theorem for subgraphs of perfect matchings

TL;DR

This paper proves an Erdős–Ko–Rado type theorem for certain subgraph families of perfect matchings, establishing optimal bounds for intersecting families and extending previous results.

Contribution

It establishes a sharp bound for the Erdős–Ko–Rado property in subgraph families of perfect matchings, improving prior bounds and confirming the conjecture for this class.

Findings

Largest intersecting family size equals sets containing a fixed vertex.

Bound n ≥ 2p + s is optimal, improving previous n ≥ 2p + 2s.

The theorem applies for all n ≥ 2p + s, confirming the Erdős–Ko–Rado property.

Abstract

Let $M_k$ be a $2n$-vertex graph with $n$ pairwise disjoint edges and let $\mathcal{H}^{(p,s)}(n)$ be the family of subsets of $V(M_n)$ that span exactly $p$ edges and $s$ isolated vertices. We prove that for $n\ge 2p+s$ this family has the Erd\H{o}s--Ko--Rado property: the size of the largest intersecting family equals to the number of sets containing a fixed vertex. The bound $n\ge 2p+s$ is the best possible, improving a recent theorem with $n\ge 2p+2s$ by Fuentes and Kamat.