Nearly-Regular Hypergraphs and Saturation of Berge Stars

Abstract

Given a graph G, we say a k-uniform hypergraph H on the same vertex set contains a Berge-G if there exists an injection φ : E(G) → E(H) such that e ⊆ φ(e) for each edge e ∈ E(G). A hypergraph H is Berge-G-saturated if H does not contain a Berge-G, but adding any edge to H creates a Berge-G. The saturation number for Berge-G, denoted sat k (n, Berge-G) is the least number of edges in a k-uniform hypergraph that is Berge-G-saturated. We determine exactly the value of the saturation numbers for Berge stars. As a tool for our main result, we also prove the existence of nearly-regular k-uniform hypergraphs, or k-uniform hypergraphs in which every vertex has degree r or r − 1 for some r ∈ Z, and less than k vertices have degree r − 1.

Saturation for the $3$-uniform loose $3$-cycle

“…Then {h , u, z} is either reasonable or rich with recipient u. By Lemma 4.9, one of the vertices v i is not a 1-flat 2-vertex, so flat(u) ≤ 2, and rich(u) + 1 2 reas(u) ≥ 1 2 . Thus, (6) holds unless flat(u) = 2, rich(u) = 0, and reas(u) = 1.…”

“…Suppose, v has at least 1 + 2j 2 j-far neighbors, and u is one of them. Let u be in k edges of the form {u, v, a}, where a is also a j-far neighbor of v. Note that by Property (1) in the definition of j-far, u is only adjacent to other j-far neighbors of v through edges that contain v. Since d(u) ≤ j, u has at most 2(j − k) neighbors in G − v. By Property (1), all these neighbors are not j-far neighbors of v, and again by Property 2, each of them is a neighbor of at most j − 1 other j-far neighbors of v. There are at least…”

“…So by the symmetry between u and w, assume without loss of generality that the core vertices of T in e are u and v, and the edge e of T contains u. But then one of the vertices in e \ {u} is in N (v), a contradiction to Property (1) in the definition of j-far. Lemma 6.2.…”

“…In terms of hypergraphs, most of the specific saturation numbers determined outside of complete graphs involve forbidden families of hypergraphs, such as triangular families [25], intersecting hypergraphs [8], and very recently Berge hypergraphs (e.g. [1,2,10,11,17]). For a detailed dynamic survey on all aspects of saturation in graphs and hypergraphs, see [14].…”

Saturation Number of Berge Stars in Random Hypergraphs

“…We say an r-uniform hypergraph H is a Berge-G if there exists a bijection φ : E(G) → E(H) such that e ⊆ φ(e) for each edge e ∈ E(G). Recently, extremal problems for Berge hypergraphs have attracted the attention of a lot of researchers, see, e.g., [3,2,4,5,8,13]. In 2018, Austhof and English [2] studied the saturation number of Berge stars.…”