The covering radius problem for sets of perfect matchings

TL;DR

This paper investigates the maximum size of collections of perfect matchings in complete graphs for which a new perfect matching exists with limited edge agreement, using probabilistic methods and the Lovász local lemma.

Contribution

It introduces new bounds for the covering radius problem of perfect matchings, extending extremal combinatorics results and proposing a conjecture for hypergraph matchings.

Findings

Derived lower bounds for the function f(n,x) using probabilistic methods.

Applied Lovász local lemma to establish bounds involving edge appearances.

Connected the problem to extremal results on permutations and hypergraph matchings.

Abstract

Consider the family of all perfect matchings of the complete graph $K_{2n}$ with $2n$ vertices. Given any collection $\mathcal M$ of perfect matchings of size $s$, there exists a maximum number $f(n,x)$ such that if $s\leq f(n,x)$, then there exists a perfect matching that agrees with each perfect matching in $\mathcal M$ in at most $x-1$ edges. We use probabilistic arguments to give several lower bounds for $f(n,x)$. We also apply the Lov\'asz local lemma to find a function $g(n,x)$ such that if each edge appears at most $g(n, x)$ times then there exists a perfect matching that agrees with each perfect matching in $\mathcal M$ in at most $x-1$ edges. This is an analogue of an extremal result vis-\'a-vis the covering radius of sets of permutations, which was studied by Cameron and Wanless (cf. \cite{cameron}), and Keevash and Ku (cf. \cite{ku}). We also conclude with a conjecture of a more general problem in hypergraph matchings.