Rainbow Perfect and Near-Perfect Matchings in Complete Graphs with Edges Colored by Circular Distance

TL;DR

This paper characterizes when rainbow perfect matchings exist in complete graphs with edges colored by circular distance and introduces a recursive algorithm to generate multiple rainbow near-perfect matchings.

Contribution

It provides a complete characterization of rainbow perfect matchings in such graphs and proposes a recursive method for generating multiple rainbow near-perfect matchings.

Findings

Rainbow perfect matchings exist iff n=8k or n=8k+2 for even n.

A recursive algorithm can generate multiple rainbow near-perfect matchings.

The study extends understanding of rainbow matchings in circular distance colorings.

Abstract

Given an edge-colored complete graph $K_n$ on $n$ vertices, a perfect (respectively, near-perfect) matching $M$ in $K_n$ with an even (respectively, odd) number of vertices is rainbow if all edges have distinct colors. In this paper, we consider an edge coloring of $K_n$ by circular distance, and we denote the resulting complete graph by $K^{\bullet}_n$. We show that when $K^{\bullet}_n$ has an even number of vertices, it contains a rainbow perfect matching if and only if $n=8k$ or $n=8k+2$, where $k$ is a nonnegative integer. In the case of an odd number of vertices, Kirkman matching is known to be a rainbow near-perfect matching in $K^{\bullet}_n$. However, real-world applications sometimes require multiple rainbow near-perfect matchings. We propose a method for using a recursive algorithm to generate multiple rainbow near-perfect matchings in $K^{\bullet}_n$.