On the Existence of Zero-Sum Perfect Matchings of Complete Graphs

TL;DR

This paper proves that in a 2-edge-coloured complete graph with equal edges of each colour, a perfect matching with an equal number of edges of each colour always exists, solving a previously posed problem.

Contribution

It establishes the existence of zero-sum perfect matchings in complete graphs with balanced 2-edge-colourings, confirming a conjecture and providing a new result in graph theory.

Findings

Existence of zero-sum perfect matchings in balanced 2-edge-coloured complete graphs

Solution to a problem posed by Caro et al.

Independent confirmation by other researchers

Abstract

In this paper, we prove that given a 2-edge-coloured complete graph $K_{4n}$ that has the same number of edges of each colour, we can always find a perfect matching with an equal number of edges of each colour. This solves a problem posed by Caro, Hansberg, Lauri, and Zarb. The problem is also independently solved by Ehard, Mohr, and Rautenbach.