Low Weight Perfect Matchings

TL;DR

This paper proves new results on the existence of perfect matchings in complete graphs with edge weightings, answering a question and strengthening previous theorems, with optimal bounds established.

Contribution

It establishes the existence of perfect matchings with specific weight properties under new conditions, solving an open question and improving prior bounds.

Findings

Existence of perfect matchings with zero total weight under certain conditions.

Improved bounds for the weight of perfect matchings in complete graphs.

Both results are proven to be optimal.

Abstract

Answering a question posed by Caro, Hansberg, Lauri, and Zarb, we show that for every positive integer $n$ and every function $\sigma\colon E(K_{4n})\to\{-1,1\}$ with $\sigma\left(E(K_{4n})\right)=0$, there is a perfect matching $M$ in $K_{4n}$ with $\sigma(M)=0$. Strengthening a result of Caro and Yuster, we show that for every positive integer $n$ and every function $\sigma\colon E(K_{4n})\to\{-1,1\}$ with $\left|\sigma\left(E(K_{4n})\right)\right|<n^2+11n+2,$ there is a perfect matching $M$ in $K_{4n}$ with $|\sigma(M)|\leq 2$. Both these results are best possible.