A uniform bound on almost colour-balanced perfect matchings in colour-balanced cliques

TL;DR

This paper establishes a uniform bound on how close a perfect matching in a colour-balanced clique can be to perfectly balanced across multiple colours, removing previous size dependencies.

Contribution

It proves the existence of a near colour-balanced perfect matching with a bound independent of the clique size, improving upon prior bounds that depended on the size parameter n.

Findings

Existence of a perfect matching with bounded imbalance independent of n

Improved upper bound of 4^{k^2} for the imbalance measure

Advances understanding of colour-balanced structures in large graphs

Abstract

An edge-colouring of a graph $G$ is said to be colour-balanced if there are equally many edges of each available colour. We are interested in finding a colour-balanced perfect matching within a colour-balanced clique $K_{2nk}$ with a palette of $k$ colours. While it is not necessarily possible to find such a perfect matching, one can ask for a perfect matching as close to colour-balanced as possible. In particular, for a colouring $c:E(K_{2nk})\rightarrow [k]$, we seek to find a perfect matching $M$ minimising $f(M) = \sum_{i=1}^k\bigl||c^{-1}(i)\cap M|-n\bigr|$. The previous best upper bound, due to Pardey and Rautenbach, was $\min f(M)\leq \mathcal{O}(k\sqrt{nk\log k})$. We remove the $n$-dependence, proving the existence of a matching $M$ with $f(M)\leq 4^{k^2}$ for all $k$.