Transversal Hamilton cycle in the hypergraph system

In this paper, we develop a sequential Hamilton framework, which is of independent interest, settling the problem proposed by Gupta et al. (2023) when k = 3, and draw the general conclusion for any k ⩾ 3 as follows. A k-graph system H = {H_i}_i∈[m] is a family of not necessarily distinct k-graphs on the same n-vertex set V; moreover, a k-graph H on V with m edges is transversal in H if there is a bijection φ: E(H) → [m] such that e ∈ E(H_φ(e)) for each e ∈ E(H). We show that given γ > 0, k ⩾ 3, sufficiently large n and an n-vertex k-graph system H = {H_i}_i∈[n], if δk−2(Hi)⩾(5/9+γ)(n2) for i ∈ [n], then there exists a tight Hamilton cycle which is transversal in H. This result implies the conclusion in a single graph, which was proved by Lang and Sanhueza-Matamala (2022) and Polcyn et al. (2021) independently.