The Study On The Existence Of S-matching In Uniform Hypergraphs

Let H=(V(H),E(H))be a hypergraph,where V(H)is the vertex set,E(H)(?)2V(H)is the edge set.If every edge e ∈ E(H)satisfies |e|=k,we call H a k-uniform hypergraph.A matching of k-uniform hypergraph is a set of pairwise disjoint edges.If the size of M is s,we call M an s-matching.M is said to be a perfect matching if M covers every vertex of H.Let deg(u)denote the degree of vertex u in H.Many open problems in combinatorics can be formulated as a problem of finding perfect matchings in hypergraphs,e.g.,Ryser ’s conjecture and the existence of combinatorial designs.In this thesis,we mainly study the existence of matching of size s.Let S1 and S2 be two(k-1)-subsets in 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.Let Hn,3,s2 denote the 3 uniform hypergraph whose vertex set is partitioned into two sets S and T of size n-2s+1 and 2s-1,respectively,and whose edge set consists of all the 3-sets with at least 2 vertices in T.The main results of this thesis are as follows:1.Let H be a 3-uniform hypergraph of order n≥13s without isolated vertices.If deg(u)+deg(v)>2((n-12)-(n-s2))for any two adjacent vertices u,v ∈ V(H),then H contains an s-matching if and only if H is not a subgraph of Hn,3,s2.2.Suppose that H is a 3-uniform hypergraph whose order n is sufficiently large and divisible by 3,if H contains no isolated vertex and deg(u)+deg(v)>2(s-1)(n-1)for any two vertices u and v that are contained in some edge of H,then H contains a matching of size The result is tight.3.Let H be a k-uniform hypergraph of order n>2k3(s+1).If |{u ∈V(H):deg(u)≤(n-1 k-1)-(n-s k-1)}|≤k-1 and deg(u)+deg(v)>2((n-1 k-1)-(n-s k-1))for any two adjacent vertices u,v ∈ V(H),then H contains an s-matching.4.Let s≥1 be an integer and H be a k-uniform hypergraph of order n≥ks+(k-2)k.If the degree sum of any two middle independent(k-1)-subsets is larger than 2(s-1),then H contains an s-matching.5.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.