Minimum degree thresholds for Hamilton $(\ell,k-\ell)$-cycles in $k$-uniform hypergraphs

Abstract

Let n > k > ℓ be positive integers. We say a k-uniform hypergraph H contains a Hamilton (ℓ, k − ℓ)-cycle if there is a partition (L0, R0, L1, R1, . . . , Lt−1, Rt−1) of V (H) with |Li| = ℓ, |Ri| = k − ℓ such that Li ∪ Ri and Ri ∪ Li+1 (subscripts module t) are all edges of H for i = 0, 1, . . . , t − 1. In the present paper, we determine the tight minimum ℓ-degree condition that guarantees the existence of a Hamilton (ℓ, k − ℓ)-cycle in every k-uniform n-vertex hypergraph for k ≥ 7, k/2 ≤ ℓ ≤ k − 1 and sufficiently large n ∈ kN.