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

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 • n . We show that when K • 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 • 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 • n . KeywordsRainbow perfect matching • Edge-colored complete graph • Circular distance • Sports scheduling • Round-robin tournament . Kostochka and Yancey [9] completed a proof of Wang and Li's conjecture. Letting n be the size of vertices of a graph G, Lo [10] showed that an edge-colored graph G contains a rainbow matching of size at least k, where k = min δ(G), 2n−4 7