Ore-type condition for Hamilton ℓ-cycle in k-uniform hypergraphs

Abstract

The classic Ore theorem states that if the degree sum of any two non-adjacent vertices in an $n$-vertex graph is at least $n$, then the graph contains a Hamilton cycle. Tang and Yan extended this result to hypergraphs in 2017 and obtained an Ore-type condition for the existence of tight Hamilton cycles. In this paper, we consider the Ore-type sufficient condition for the existence of Hamilton $\ell$-cycles. Our main result is that for $k\ge 3$, $1\le \ell <k/2$, $\gamma >0$ and for sufficiently large $n$ with $n\in (k-\ell )N$, every $k$-uniform hypergraph $H=(V,E)$ on $n$ vertices with the degree sum of any two weakly independent sets at least $(\frac{3}{2(k-\ell )}+\gamma )n$ contains a Hamilton $\ell$-cycle.