Almost color-balanced perfect matchings in color-balanced complete graphs

TL;DR

This paper investigates the existence of near color-balanced perfect matchings in complete graphs with edge colorings, providing bounds on how close such matchings can be to perfectly balanced, especially for specific numbers of colors.

Contribution

It proves the existence of near color-balanced perfect matchings with bounded deviation in complete graphs for general and specific color counts, advancing understanding of color-balanced structures.

Findings

Existence of perfect matchings with deviation O(k√kn ln(k)) for general k.

Existence of perfectly color-balanced matchings when k=3.

Prior results for k=2 are extended to larger k.

Abstract

For a graph $G$ and a not necessarily proper $k$-edge coloring $c:E(G)\to \{ 1,\ldots,k\}$, let $m_i(G)$ be the number of edges of $G$ of color $i$, and call $G$ {\it color-balanced} if $m_i(G)=m_j(G)$ for every two colors $i$ and $j$. Several famous open problems relate to this notion; Ryser's conjecture on transversals in latin squares, for instance, is equivalent to the statement that every properly $n$-edge colored complete bipartite graph $K_{n,n}$ has a color-balanced perfect matching. We contribute some results on the question posed by Kittipassorn and Sinsap (arXiv:2011.00862v1) whether every $k$-edge colored color-balanced complete graph $K_{2kn}$ has a color-balanced perfect matching $M$. For a perfect matching $M$ of $K_{2kn}$, a natural measure for the total deviation of $M$ from being color-balanced is $f(M)=\sum\limits_{i=1}^k|m_i(M)-n|$. While not every color-balanced complete graph $K_{2kn}$ has a color-balanced perfect matching $M$, that is, a perfect matching with $f(M)=0$, we prove the existence of a perfect matching $M$ with $f(M)=O\left(k\sqrt{kn\ln(k)}\right)$ for general $k$ and $f(M)\leq 2$ for $k=3$; the case $k=2$ has already been studied earlier. An attractive feature of the problem is that it naturally invites the combination of a combinatorial approach based on counting and local exchange arguments with probabilistic and geometric arguments.