Colour-balanced subgraphs

TL;DR

This paper proves that colour-balanced complete graphs contain nearly colour-balanced perfect matchings, improving bounds for various classes of subgraphs and hypergraphs using linear relaxations and necklace-splitting.

Contribution

It resolves a conjecture by showing the existence of nearly colour-balanced perfect matchings in colour-balanced graphs with improved bounds.

Findings

Any colour-balanced $k$-edge-coloured complete graph $K_{2kt}$ contains a nearly colour-balanced perfect matching.

The results extend to bounded-degree spanning subgraphs and hypergraphs with improved bounds.

The proofs utilize linear relaxations and necklace-splitting techniques to establish the bounds.

Abstract

A $k$-edge-coloured graph is colour-balanced if each colour appears equally often. Resolving a conjecture of Pardey and Rautenbach, we show that any colour-balanced $k$-edge-coloured complete graph $K_{2kt}$ contains a perfect matching that can be made colour-balanced by recolouring $O(k^2)$ edges. More generally, we obtain analogous bounds for arbitrary bounded-degree spanning subgraphs of edge-coloured complete graphs and for perfect matchings in edge-coloured $r$-uniform complete hypergraphs in a more general vector-label setting. The former result answers a question recently posed by Banerjee and Hollom, and significantly improves earlier bounds for all previously studied classes of subgraph. Our proofs reduce each of these problems to a setting in which we can apply a bound for perfect matchings in the complete bipartite graph, established via a linear relaxation and a necklace-splitting argument.