Let SI 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≠ 0 and S2 ∩ e ≠0 or e S1 ∪...Let SI 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≠ 0 and S2 ∩ e ≠0 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.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.11771247)
文摘Let SI 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≠ 0 and S2 ∩ e ≠0 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.
