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PreprintCC BY-NC-SA 4.0arXiv

Asymptotic Enumeration of F-decompositions in Typical Hypergraphs and of Universal Cycles

  • Danni Li
  • Bin Wang
  • Junxue Zhang
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Abstract

In this paper, we determine the asymptotic number of F-decompositions in typical hypergraphs. More precisely, let F be a non-trivial f-vertex k-graph. We show that, for all sufficiently large n, every n-vertex F-divisible (n−ζ,h,p)-typical k-graph G has n(f−k+on(1))/|F||G| F-decompositions. As an application, we asymptotically enumerate universal cycles for k-subsets of [n], or equivalently, tight Euler tours of the complete k-graph Knk. For every k≥2 and all sufficiently large n satisfying k| n−1 choose k−1, the number of such universal cycles is n(1−γ(n))n choose k, where 0<γ(n)=on(1). More generally, for ℓ∈[k−1], we show that a sufficiently typical n-vertex k-graph G satisfying the corresponding divisibility condition has n(k−ℓ−γ'(n))|G| Euler ℓ-tours, where 0<γ'(n)=on(1).

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Cite

BibTeXRIS
BibTeX
@misc{li2026asymptotic,
  title = {{Asymptotic Enumeration of $F$-decompositions in Typical Hypergraphs and of Universal Cycles}},
  author = {Danni Li and Bin Wang and Junxue Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26579},
  eprint = {2610.26579},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26579},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Asymptotic Enumeration of F-decompositions in Typical Hypergraphs and of Universal Cycles
AU  - Danni Li
AU  - Bin Wang
AU  - Junxue Zhang
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26579
UR  - https://arxiv.org/abs/2610.26579
AB  - In this paper, we determine the asymptotic number of F-decompositions in typical hypergraphs. More precisely, let F be a non-trivial f-vertex k-graph. We show that, for all sufficiently large n, every n-vertex F-divisible (n^−ζ,h,p)-typical k-graph G has n^(f−k+o_n(1))/|F||G| F-decompositions. As an application, we asymptotically enumerate universal cycles for k-subsets of [n], or equivalently, tight Euler tours of the complete k-graph K_n^k. For every k≥2 and all sufficiently large n satisfying k| n−1 choose k−1, the number of such universal cycles is n^(1−γ(n))n choose k, where 0<γ(n)=o_n(1). More generally, for ℓ∈[k−1], we show that a sufficiently typical n-vertex k-graph G satisfying the corresponding divisibility condition has n^(k−ℓ−γ'(n))|G| Euler ℓ-tours, where 0<γ'(n)=o_n(1).
ER  -