Abstract
A hypergraph H = ( V , ε ) is a pair consisting of a vertex set V , and a set ε of subsets (the hyperedges of H ) of V . A hypergraph H is r -uniform if all the hyperedges of H have the same cardinality r . Let H be an r -uniform hypergraph, we generalize the concept of trees for r -uniform hypergraphs. We say that an r -uniform hypergraph H is a generalized hypertree ( G H T ) if H is disconnected after removing any hyperedge E , and the number of components of G H T − E is a fixed value k ( 2 ≤ k ≤ r ) . We focus on the case that G H T − E has exactly two components. An edge-minimal G H T is a G H T whose edge set is minimal with respect to inclusion. After considering these definitions, we show that an r -uniform G H T on n vertices has at least 2 n / ( r + 1 ) edges and it has at most n − r + 1 edges if r ≥ 3 and n ≥ 3 , and the lower and upper bounds on the edge number are sharp. We then discuss the case that G H T − E has exactly k ( 2 ≤ k ≤ r − 1 ) components.
Highlights
Graph theory is a powerful tool in exploring various network types, and in many areas of application, such as communication, biology, and chemistry
We further develop the concept proposed by Nieminen into the generalized hypertrees (GHT)
It turns out that the lower bound in Theorem 3 can be improved for GHT − E which has exactly k components for any edge E ∈ ε( GHT ) if 2 ≤ k ≤ r − 1
Summary
Graph theory is a powerful tool in exploring various network types, and in many areas of application, such as communication, biology, and chemistry. We generalize the concept of trees for hypergraphs and prove bounds on the edge numbers of such hypergraphs. Katona et al [9] discussed bounds on the edge number in another interesting generalization of trees called hypertrees. It is well-known from elementary graph theory that any two vertices in a tree are connected by a unique path. This description of the definition of a tree can reflect the edge-connectivity property of the tree and means that the tree would be disconnected if any edge is deleted from the tree. The notion of edge‐connectivity λ(G) of a graph G , defined as the minimum number of edges
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have