A note on minimum degree condition for Hamilton (a,b)-cycles in hypergraphs

Let k, a, b be positive integers with a + b = k. A k-uniform hypergraph is called an ( a, b )-cycle if there is a partition ( A 0, B 0, A 1, B 1, …, A t − 1, B t − 1 ) of the vertex set with | A i | = a, | B i | = b such that A i ∪ B i and B i ∪ A i + 1 (subscripts module t) are edges for all i = 0, 1, …, t − 1. Let H be a k-uniform n-vertex hypergraph with n ≥ 5 k and n divisible by k. By applying the concentration inequality for intersections of a uniform hypergraph with a random matching developed by Frankl and Kupavskii, we show that if there exists α ∈ ( 0, 1 ) such that δ a ( H ) ≥ ( α + o ( 1 ) ) ( n − a b ) and δ b ( H ) ≥ ( 1 − α + o ( 1 ) ) ( n − b a ), then H contains a Hamilton ( a, b )-cycle. As a corollary, we prove that if δ ℓ ( H ) ≥ ( 1 / 2 + o ( 1 ) ) ( n − ℓ k − ℓ ) for some ℓ ≥ k / 2, then H contains a Hamilton ( k − ℓ, ℓ )-cycle and this is asymptotically best possible.