A k-edge-coloured graph is balanced if each colour appears equally often. 

Given a balanced k-edge-colouring of a complete graph with an even number of vertices, is there a perfect matching on which the colouring is close to balanced? Proving a strengthened version of a conjecture of Pardey and Rautenbach, we show that there is a perfect matching that is within O(k^2) edges of balanced. More generally, we prove results for arbitrary bounded-degree spanning subgraphs, answering a question of Banerjee and Hollom, and extend our results to hypergraphs. This significantly improves earlier bounds for all previously studied classes of subgraph. 

Joint work with Emma Hogan and Dmitry Tsarev.