Some Ore-type Results for Matching and Perfect Matching in k-uniform Hypergraphs

Let S1 and S2 be two (k . 1)-subsets in a k-uniform hypergraph H. We call S1 and S2 strongly or middle or weakly independent if H does not contain an edge e . E(H) such that S1 . e . . and S2 . e . . or e . S1 . S2 or e . S1 . S2, respectively. In this paper, we obtain the following results concerning these three independence. (1) For any n . 2k2 . k and k . 3, there exists an n-vertex k-uniform hypergraph, which has degree sum of any two strongly independent (k . 1)-sets equal to 2n.4(k.1), contains no perfect matching; (2) Let d . 1 be an integer and H be a k-uniform hypergraph of order n . kd+(k.2)k. If the degree sum of any two middle independent (k.1)-subsets is larger than 2(d.1), then H contains a d-matching; (3) For all k . 3 and sufficiently large n divisible by k, we completely determine the minimum degree sum of two weakly independent (k . 1)-subsets that ensures a perfect matching in a k-uniform hypergraph H of order n.