Rainbow Berge Hamiltonicity in edge-colored random k-uniform hypergraphs

@misc{zhang2026rainbow,
  title = {{Rainbow Berge Hamiltonicity in edge-colored random $k$-uniform hypergraphs}},
  author = {Liping Zhang and Ailian Chen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01989},
  eprint = {2609.01989},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01989},
}

TY  - UNPB
TI  - Rainbow Berge Hamiltonicity in edge-colored random k-uniform hypergraphs
AU  - Liping Zhang
AU  - Ailian Chen
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.01989
UR  - https://arxiv.org/abs/2609.01989
AB  - Let H ∼ H^k_c(n,p) be an edge-colored random k-uniform hypergraph on the vertex set [n], where each edge e ∈ binom[n]k is included independently with probability p and is uniformly and independently assigned a color from the color set [c]. For k = 2, Ferber and Krivelevich (2016) established that if c = (1+o(1))n and p = (log n + log log n + ω(n))/n, then with high probability the edge-colored random graph H ∼ H^2_c(n,p) contains a rainbow Hamilton Berge cycle. Subsequently, Bal, Berkowitz, Devlin, and Schacht (2021) determined the threshold for the appearance of a (non-rainbow) Hamilton Berge cycle in random k-uniform hypergraphs. In this paper, we generalize the results to all integers k ≥ 3. We prove that if c = (1+o(1))n and p = (k−1)! (log n + loglog n + ω(n))/(n^k−1), then with high probability H ∼ H^k_c(n,p) contains a rainbow Hamilton Berge cycle. Furthermore, both conditions on c and p are asymptotically tight. \noindent\emph{Key words:} Rainbow subgraph, Hamiltonicity, Berge cycle, Random hypergraph.
ER  -