A Note on Minimum Degree Condition for Hamiltonian $(a,b)$-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 $(A_0,B_0,A_1,B_1,\dots ,A_{t-1},B_{t-1})$ of the vertex set with $\mathrm{\mid }{A}_i\mathrm{\mid }=a$, $\mathrm{\mid }{B}_i\mathrm{\mid }=b$ such that $A_i\cup B_i$ and $B_i\cup A_{i+1}$ (subscripts module $t$) are edges for all $i=0,1,\dots ,t-1$. Let $\mathcal{H}$ be a $k$-uniform $n$-vertex hypergraph with $n\ge 5k$ 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 $\alpha \in (0,1)$ such that $\delta_a($\mathcal{H}$)\geq (\alpha +o(1))\binom{n-a}{b}$and$\delta_b($\mathcal{H}$)\geq (1-\alpha +o(1))\binom{n-b}{a}$,then$$\mathcal{H}$$containsaHamilton$(a,b)$-cycle. As a corollary, we prove that if $\delta_{\ell}($\mathcal{H}$)\geq (1/2 +o(1))\binom{n-\ell}{k-\ell}$forsome$\ell \geq k/2$,then$$\mathcal{H}$$containsaHamilton$(k-\ell,\ell)$-cycle and this is asymptotically best possible.