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arXiv · 2604.09449

Colour-balanced subgraphs

Emma Hogan·Alex Scott·Dmitry Tsarev
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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.

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Emma Hogan, Alex Scott, Dmitry Tsarev. 2026-04-10. Colour-balanced subgraphs. https://arxiv.org/abs/2604.09449

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