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

Authors

Shuhei Saitoh, Seikei UniversityFollow
Naoki Matsumoto, Keio UniversityFollow
Wei Wu, Shizuoka UniversityFollow

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.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Saitoh, Shuhei; Matsumoto, Naoki; and Wu, Wei (2022) "Rainbow Perfect and Near-Perfect Matchings in Complete Graphs with Edges Colored by Circular Distance," Theory and Applications of Graphs: Vol. 9: Iss. 1, Article 9.
DOI: 10.20429/tag.2022.090109
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol9/iss1/9