Uniformly balanced $H$-factors in multicoloured complete graphs
Agnijo Banerjee, Lawrence Hollom
TL;DR
The paper resolves Hollom's question by proving that for fixed integers $k,r\ge 2$ there exists a constant $C_{k,r}$ such that every graph $H$ on $r$ vertices and every balanced $k$-colouring of $K_{nrk}$ contains an $H$-factor $F$ whose colour distribution deviates from perfect balance by at most $C_{k,r}$. It generalises to colourings with colours as unit vectors in $\mathbb{S}^{d-1}$, yielding a bound on $\|\sum_{e\in E(F)} c(e)\|$; in the simplex case an explicit bound $C=(8dr)^{(8dr)^{d+1}-1}$ is given. The proof proceeds in two stages: first construct a colour-balanced $K_r$-factor via a contradictory-swaps argument and a geometric hyperplane partition, then embed an $H$-factor inside the $K_r$-factor to inherit the balance up to a fixed additive error. The results unify discrete and continuous colour models and suggest directions for tightening constants and extending to broader graph families.
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.