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

Abstract

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.