Uniformly balanced $H$-factors in multicoloured complete graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915756493504512 |
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| author | Banerjee, Agnijo Hollom, Lawrence |
| author_facet | Banerjee, Agnijo Hollom, Lawrence |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18789 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniformly balanced $H$-factors in multicoloured complete graphs Banerjee, Agnijo Hollom, Lawrence Combinatorics 05C15, 05C70 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. |
| title | Uniformly balanced $H$-factors in multicoloured complete graphs |
| topic | Combinatorics 05C15, 05C70 |
| url | https://arxiv.org/abs/2601.18789 |