Uniformly balanced $H$-factors in multicoloured complete graphs

TL;DR

This paper proves that in multicoloured complete graphs, one can find an $H$-factor with nearly balanced colour distribution, extending previous results to more general colour palettes and providing bounds on imbalance.

Contribution

The authors establish the existence of an $H$-factor with bounded imbalance in multicoloured complete graphs, generalizing prior results to continuous colour palettes.

Findings

Existence of an $H$-factor with at most $C_{r,k}$ edges imbalance.

Extension to colour palettes in $ extbf{S}^{d-1}$ with bounded Euclidean norm.

Answer to Hollom's question for general colourings and graphs.

Abstract

A balanced colouring of a graph is one in which every colour appears the same number of times. Given a fixed graph $H$ on $r$ vertices and a balanced $k$-colouring of the complete graph $K_{nrk}$, Hollom (2025) asked the following question: can we always find an $H$-factor $F$ covering all vertices of the complete graph $K_{nrk}$ such that the inherited colouring of $F$ is almost balanced? This is known to be the case for palettes of only two colours, or when $H$ is only a single edge. We answer the above question in full, finding an $H$-factor which is at most $C_{r,k}$ edges away from being balanced, where $C_{r,k}$ depends only on $r$ and $k$. In fact, we work in the more general setting wherein our palette of colours is a subset of $\mathbb{S}^{d-1}$, and find an $H$-factor where the sum of the colours of all edges has bounded Euclidean norm.