Hamilton-chain saturated hypergraphs

AbstractWe say that a hypergraph H is hamiltonian path (cycle) saturated if H does not contain an open (closed) hamiltonian chain but by adding any new edge we create an open (closed) hamiltonian chain in H. In this paper we ask about the smallest size of an r-uniform hamiltonian path (cycle) saturated hypergraph, mainly for r=3. We present a construction of a family of 3-uniform path (cycle) saturated hamiltonian hypergraphs with O(n5/2) edges. On the other hand we prove that the number of edges in an r-uniform hamiltonian path (cycle) saturated hypergraph is at least Ω(nr−1)

Exact Bounds for Some Hypergraph Saturation Problems

Let W_n(p,q) denote the minimum number of edges in an n x n bipartite graph G on vertex sets X,Y that satisfies the following condition; one can add the edges between X and Y that do not belong to G one after the other so that whenever a new edge is added, a new copy of K_{p,q} is created. The problem of bounding W_n(p,q), and its natural hypergraph generalization, was introduced by Balogh, Bollob\'as, Morris and Riordan. Their main result, specialized to graphs, used algebraic methods to determine W_n(1,q). Our main results in this paper give exact bounds for W_n(p,q), its hypergraph analogue, as well as for a new variant of Bollob\'as's Two Families theorem. In particular, we completely determine W_n(p,q), showing that if 1 <= p <= q <= n then W_n(p,q) = n^2 - (n-p+1)^2 + (q-p)^2. Our proof applies a reduction to a multi-partite version of the Two Families theorem obtained by Alon. While the reduction is combinatorial, the main idea behind it is algebraic

Bounding the Number of Hyperedges in Friendship rr-Hypergraphs

For $r \ge 2$, an $r$-uniform hypergraph is called a friendship $r$-hypergraph if every set $R$ of $r$ vertices has a unique 'friend' - that is, there exists a unique vertex $x \notin R$ with the property that for each subset $A \subseteq R$ of size $r-1$, the set $A \cup \{x\}$ is a hyperedge. We show that for $r \geq 3$, the number of hyperedges in a friendship $r$-hypergraph is at least $\frac{r+1}{r} \binom{n-1}{r-1}$, and we characterise those hypergraphs which achieve this bound. This generalises a result given by Li and van Rees in the case when $r = 3$. We also obtain a new upper bound on the number of hyperedges in a friendship $r$-hypergraph, which improves on a known bound given by Li, van Rees, Seo and Singhi when $r=3$.Comment: 14 page

Not All Saturated 3-Forests Are Tight

A basic statement in graph theory is that every inclusion-maximal forest is connected, i.e. a tree. Using a definiton for higher dimensional forests by Graham and Lovasz and the connectivity-related notion of tightness for hypergraphs introduced by Arocha, Bracho and Neumann-Lara in, we provide an example of a saturated, i.e. inclusion-maximal 3-forest that is not tight. This resolves an open problem posed by Strausz