On Non-Uniform Baranyai Factorizations
Abstract
For a fixed nonempty finite set L of positive integers and k=max L≥2, let binom[n]L=⋃ℓ∈ Lbinom[n]ℓ denote the non-uniform hypergraph whose edge set consists of all ℓ-subsets of [n] for ℓ∈ L. A $1$-factor of a hypergraph H is a collection of pairwise disjoint edges whose union is V(H). We say H is $1$-factorable if E(H) can be partitioned into $1$-factors. In this paper, we study the 1-factorability of binom[n]L. Earlier work of He, Huang, and Ma shows that, for all sufficiently large n, $1$-factorability can occur only in the residue classes $0$ and −1 modulo k: it always occurs in class $0$, whereas it occurs in class −1 only if k−1∈ L. We determine exactly when $1$-factorability occurs in this exceptional class. More precisely, we define a finite gcd-peeling algorithm on L∖{k−1,k} and prove that binom[qk−1]L is $1$-factorable if and only if the algorithm accepts. As a consequence, our result provides a criterion for the $1$-factorability of binom[n]L and confirms the conjecture posed by He, Huang, and Ma.
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BibTeX
@misc{gao2026nonuniform,
title = {{On Non-Uniform Baranyai Factorizations}},
author = {Guorong Gao and Mingze Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24695},
eprint = {2610.24695},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24695},
}
RIS
TY - UNPB
TI - On Non-Uniform Baranyai Factorizations
AU - Guorong Gao
AU - Mingze Li
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24695
UR - https://arxiv.org/abs/2610.24695
AB - For a fixed nonempty finite set L of positive integers and k=max L≥2, let binom[n]L=⋃_ℓ∈ Lbinom[n]ℓ denote the non-uniform hypergraph whose edge set consists of all ℓ-subsets of [n] for ℓ∈ L. A $1$-factor of a hypergraph H is a collection of pairwise disjoint edges whose union is V(H). We say H is $1$-factorable if E(H) can be partitioned into $1$-factors. In this paper, we study the 1-factorability of binom[n]L. Earlier work of He, Huang, and Ma shows that, for all sufficiently large n, $1$-factorability can occur only in the residue classes $0$ and −1 modulo k: it always occurs in class $0$, whereas it occurs in class −1 only if k−1∈ L. We determine exactly when $1$-factorability occurs in this exceptional class. More precisely, we define a finite gcd-peeling algorithm on L∖{k−1,k} and prove that binom[qk−1]L is $1$-factorable if and only if the algorithm accepts. As a consequence, our result provides a criterion for the $1$-factorability of binom[n]L and confirms the conjecture posed by He, Huang, and Ma.
ER -