On saturation of Berge hypergraphs

A hypergraph $H=(V(H), E(H))$ is a Berge copy of a graph $F$, if $V(F)\subset V(H)$ and there is a bijection $f:E(F)\rightarrow E(H)$ such that for any $e\in E(F)$ we have $e\subset f(e)$. A hypergraph is Berge-$F$-free if it does not contain any Berge copies of $F$. We address the saturation problem concerning Berge-$F$-free hypergraphs, i.e., what is the minimum number $sat_r(n,F)$ of hyperedges in an $r$-uniform Berge-$F$-free hypergraph $H$ with the property that adding any new hyperedge to $H$ creates a Berge copy of $F$. We prove that $sat_r(n,F)$ grows linearly in $n$ if $F$ is either complete multipartite or it possesses the following property: if $d_1\le d_2\le \dots \le d_{|V(F)|}$ is the degree sequence of $F$, then $F$ contains two adjacent vertices $u,v$ with $d_F(u)=d_1$, $d_F(v)=d_2$. In particular, the Berge-saturation number of regular graphs grows linearly in $n$.


Introduction
Given a family F of (hyper)graphs, we say that a (hyper)graph G is F -free if G does not contain any member of F as a subhypergraph. The obvious question is how large an F -free (hyper)graph can be, i.e. what is the maximum number ex(n, F ) of (hyper)edges in an F -free n-vertex (hyper)graph is called the extremal/Turán problem. A natural counterpart to this well-studied problem is the so-called saturation problem. We say that G is F -saturated if G is F -free, but adding any (hyper)edge to G creates a member of F . The question is how small an F -saturated (hyper)graph can be, i.e. what is the minimum number sat(n, F ) of (hyper)edges in an F -saturated n-vertex (hyper)graph.
In the graph case, the study of saturation number was initiated by Erdős, Hajnal, and Moon [5]. Their theorem on complete graphs was generalized to complete uniform hypergraphs by Bollobás [2]. Kászonyi and Tuza [9] showed that for any family F of graphs, we have sat(n, F ) = O(n). For hypergraphs, Pikhurko [10] proved the analogous result that for any family F of r-uniform hypergraphs, he proved that we have sat(n, F ) = O(n r−1 ).
For some further types of saturation ("strongly F -saturated" and "weakly F -saturated" hypergraphs) the exact exponent of n was determined in [11] for every forbidden hypergraph F .
In this paper, we consider some special families of hypergraphs. We say that a hypergraph H is a Berge copy of a graph F (in short: H is a Berge-F ) if V (F ) ⊂ V (H) and there is a bijection f : E(F ) → E(H) such that for any e ∈ E(F ) we have e ⊂ f (e). We say that F is a core graph of H. Note that there might be multiple core graphs of a Berge-F hypergraph and F might be the core graph of multiple Berge-F hypergraphs.
Berge hypergraphs were introduced by Gerbner and Palmer [8], extending the notion of hypergraph cycles in Berge's definition [1]. They studied the largest number of hyperedges in Berge-F -free hypergraphs (and also the largest total size, i.e. the sum of the sizes of the hyperedges). English, Graber, Kirkpatrick, Methuku and Sullivan [3] considered the saturation problem for Berge hypergraphs. They conjectured that sat r (n, Berge-F ) = O(n) holds for any r and F , and proved it for several classes of graphs. Here and throughout the paper the parameter r in the index denotes that we consider r-uniform hypergraphs, and we will denote sat r (n, Berge-F ) by sat r (n, F ) for brevity. The conjecture was proved for 3 ≤ r ≤ 5 and any F in [4]. In this paper we gather some further results that support the conjecture.
English, Gerbner, Methuku and Tait [4] extended this conjecture to hypergraph-based Berge hypergraphs. Analogously to the graph-based case, we say that a hypergraph H is a Berge copy of a hypergraph F (in short: H is a Berge-F ) if V (F ) ⊂ V (H) and there is a bijection f : E(F ) → E(H) such that for any e ∈ E(F ) we have e ⊂ f (e). We say that F is a core hypergraph of H. The conjecture in this case states that if F is a u-uniform hypergraph, then sat r (n, F ) = O(n u−1 ).
For a hypergraph H = (V (H), E(H)) and a family of hypergraphs F we say that H is F -oversaturated if for any hyperedge h ⊂ V (H) that is not in H, there is a copy of a hypergraph F ∈ F that consists of h and |E(F )| − 1 hyperedges in E(H). Let osat r (n, F ) denote the smallest number of hyperedges in an F -oversaturated r-uniform hypergraph on n vertices. Proposition 1.1. For any u-uniform hypergraph F and any r > u, we have osat r (n, F ) = O(n u−1 ). Moreover, there is an r-uniform hypergraph H with O(n u−1 ) hyperedges such that adding any hyperedge to H creates a Berge-F such that its core hypergraph F 0 (which is a copy of F ) is not a core hypergraph of any Berge-F in H.
We remark that in the case u = 2, the linearity of osat r (n, F ) follows from either of the next two theorems, as they imply sat r (n, K k ) = O(n). Indeed, if v is the number of vertices of any graph in F , then any Berge-K v -saturated hypergraph is obviously Berge-Foversaturated.
Let F be a fixed graph on v vertices with degree sequence d 1 ≤ d 2 ≤ · · · ≤ d v . Set δ := d 2 − 1. We say that F is of type I if there exist vertices u 1 , u 2 with d F (u 1 ) = d 1 , d F (u 2 ) = d 2 that are joined with an edge. Otherwise F is called of type II. Observe that any regular graph is of type I.

Proofs
Proof of Proposition 1.1. We define V (H) as the disjoint union of a set R of size r − u and a set L of size n − r + u. We take an F -saturated u-uniform hypergraph G with vertex set L that contains O(n u−1 ) hyperedges. Such a hypergraph exists by the celebrated result of Pikhurko [10]. As another alternative, one may take an oversaturated hypergraph with O(n u−1 ) hyperedges, whose existence is guaranteed by [11,Theorem 1]. Then we let the hyperedges h of H be the r-sets with the property that h ∩ L is a hyperedge of G or has at most u − 1 vertices from L.
Obviously H has O(n u−1 ) hyeredges. Clearly we have that every r-set h that is not a hyperedge of H contains a u-element subset e of L that is not a hyperedge of G. Then e creates a copy of F . We let f (e) = h, and for each other edge e ′ of that copy of F , we let f (e ′ ) = e ′ ∪ R. This shows that this copy of F is the core of a Berge-F .
Proof of Theorem 1.2. We consider two cases according to how large the uniformity r is compared to the sum of class sizes k 1 , k 2 , . . . , k s+1 . We set N := s i=1 k i − 1. For brevity, we write K for K k 1 ,k 2 ,...,k s+1 .
Indeed, a copy of a Berge-K must contain a vertex v in the smallest s classes of the core from outside C. But then, if v ∈ B i , either the whole copy is in C ∪ B i or C must contain all classes of the core of the copy. As none of these are possible, G is indeed Berge-K-free.
Next observe that adding any r-set G to G that contains two vertices u and v from different This shows that the additional hyperedges of any K-saturated family that contains G are subsets of C ∪ B i ∪ R for some i, and hence there is only a linear number of them. As G also contains a linear number of hyperedges, the total size of such K-saturated families is O(n).
. , x N be the elements of C, let e 1 , e 2 , . . . , e ( N 2 ) be the edges of the complete graph on C, and finally let π 1 , π 2 , . . . , π ( N 2 ) be permutations of C such that the endvertices of e i are the values π i (j), π i (j + 1) for some j.
Then let us define the family G as First, we claim that G is Berge-K-free. Indeed, there are only N vertices with degree at least N + 1.
Next, observe that if we add a family F to G that contains a Berge- if v is the center of the star, then C ∪ {v} plays the role of the smallest classes of K, and k s+1 leaves that belong to distinct B i s can play the role of the largest class of K. Here, we use the facts that every edge in C 2 is contained in an unbounded number of hyperedges of G as n tends to infinity and that for any vertex u ∈ ∪ m i=1 B i , G contains a Berge-star with center u and core C ∪ {u}; and if u and u ′ belong to different B i 's, then the hyperedges of these Berge-stars are distinct.
Let F be such that F ∪ G is Berge-K-free. Then by the above, finishes the proof. It is well-known that forbidding a Berge-star (or any Berge-tree) results in O(n) hyperedges, but for sake of completeness we include a proof for stars.
Observe that F ′ being Berge-S k s+1 (r−2) -free is equivalent to the condition that for every x ∈ ∪ m i=1 B i the family {F \ {x} : x ∈ F ∈ F ′ } is not disjointly k s+1 (r − 2)-representable, i.e. there do not exist y 1 , y 2 , . . . , y k s+1 (r−2) and sets F 1 , F 2 , . . . , j=1,j =α F j . By a well-known result of Frankl and Pach [7] if all sets of a family H with this property have size at most r, then |H| is bounded by a constant depending only on r and k s+1 (r − 2). That is, |F ′ x | is bounded by the same constant independently of x, and therefore the size of F ′ , and thus the size of G is linear. We obtained that any k-saturated family G ′ with G ⊂ G ′ has O(n) hyperedges.
Proposition 2.1. Let F be a graph with no isolated vertex and with an isolated edge (u 1 , u 2 ). Then for any r ≥ 3 we have sat r (n, F ) = O(n).
Proof. Let U be a set of size n. Let v denote the number of vertices of F , let F ′ be the graph obtained from F by removing the edge (u 1 , u 2 ) and let C be a (v − 2)-subset of U. Suppose first r ≤ v − 1, and let G 0 be a Berge copy of F ′ with core C and G 0 ⊆ G C,1 ⊆ U r , where G C,1 is the set of r-sets that contain at most one vertex from U \ C. Note that G C,1 contains a linear number of r-subsets. Then let G be an r-graph with G 0 ⊆ G ⊆ G C,1 such that any H ∈ G C,1 \ G creates a Berge copy of F with G. Then G has linearly many hyperedges and is clearly F -saturated since if G contains at least two vertices from U \ C, then G can play the role of (u 1 , u 2 ) and together with the Berge copy of F ′ they form a Berge-F .
If r ≥ v, then any G with e(F ) − 1 r-subsets sharing v − 2 common elements (denote their set by C) is F -saturated. Indeed, any additional r-set G contains at least 2 vertices not in C, so those two vertices can play the role of u 1 and u 2 , G can play the role of the edge (u 1 , u 2 ), and the r-sets of G form a Berge copy of F ′ with core C.
Observe that if F is of type I, then it cannot contain isolated vertices, and since graphs with an isolated edge are covered by Proposition 2.1, we may and will assume that d 2 − 1 = δ ≥ 1 holds.
Proof of Theorem 1.3. Let F be a graph of type I on v vertices and let u 1 , u 2 be a pair of vertices of F showing the type I property. Set d := |N(u 1 ) ∩ N(u 2 )| and let F ′ denote the subgraph of F on N(u 1 ) ∩ N(u 2 ) spanned by the edges incident to u 1 or u 2 with the edge (u 1 , u 2 ) removed. Our strategy to prove the theorem is to construct a Berge-F -free r-graph G with O(n) hyperedges such that any F -saturated r-graph G ′ ⊃ G contains at most a linear number of extra hyperedges.
Let us say that G is F -good if its vertex set V can be partitioned into V = C ∪ B 1 ∪ B 2 ∪ · · · ∪ B m ∪ R such that |C| = v − 2, all B i 's have equal size b at most r, |R| < b and the following properties hold: 1. every hyperedge of G not contained in C is of the form A ∪ B i for some i = 1, 2, . . . , m with A ⊂ C, 3. for every 1 ≤ i < j ≤ m and y ∈ B i , y ′ ∈ B j , the sub-r-graph {G ∈ G : y ∈ G} ∪ {G ∈ G : y ′ ∈ G} contains a Berge-F ′ with y, y ′ being the only vertices of the core not in C and y, y ′ playing the role of u 1 , u 2 , 2 ) is an enumeration of the edges of the complete graph on C, then e h ⊂ G h for all h = 1, 2, . . . , v−2 2 , i.e., these hyperedges form a Berge-K v−2 with core C. Claim 2.2. If G is F -good, then G is Berge-F -free and any F -saturated supergraph G ′ of G contains at most a linear number of extra edges compared to G.
Proof of Claim. Observe first that G is Berge-F -free as the core of a copy of a Berge-F should contain at least two vertices not in C, both of degree δ < d 2 .
Next, we claim that for any hyperedge H meeting two distinct B's, say B i and B j , the r-graph G ∪ {H} contains a Berge-F . Indeed, let y ∈ B i ∩ H, y ′ ∈ B j ∩ H. Then by item 3 of the F -good property, y can play the role of u 1 and y ′ can play the role of u 2 , H can play the role of the edge (u 1 u 2 ), and item 4 of the F -good property ensures that the other vertices of C can play the role of the rest of the core of F .
Finally, let G ′ be any F -saturated r-graph containing G. Then by the above, any hyperedge in G ′ \ G meets at most one B i , and thus is of the form P ∪ Q with P ⊂ C ∪ R, Q ⊂ B i for some i. The number of such sets is at most 2 b 2 v−2+b m = O(n). Claim 2.3. For any type I graph F on v ≥ 7 vertices with δ > 0 and any integer r ≥ 6 there exists an F -good r-graph G with O(n) hyperedges.
Proof of Claim. We fix a set D ⊂ C of size d.
Then putting all r-subsets of C into G ensures item 4 of the F -good property. We set b = r−2, so all further sets will meet C in 2 vertices. Observe that . , x d }, and y 1,1 , y 1,2 , y 2,1 , y 2,2 , . . . , where addition is always modulo the underlying set, i.e., G 0 consists of two cycles and a matching. Let us put all sets of the form A ∪ B h with A ∈ G 0 and 1 ≤ h ≤ m into G. Then items 1 and 2 of the F -good property are satisfied, thus we need to check item 3.
Case II. r > v − 4 Then we set b = r − (v − 4) and thus every hyperedge meets C in c := v − 4 = |C| − 2 vertices. Consequently, |R| is the residue of n − v + 2 modulo b. By v ≥ 7, we obtain c ≥ 3. Let e 1 , e 2 , . . . , e ( v−2 2 ) be an enumeration of the edges of the complete graph on C. Then for any 1 ≤ h ≤ m, we will put a hyperedge of the form 2 ). As n tends to infinity, so does m, and this will ensure item 4 of the F -good property.
Suppose first d > 0 and observe that ( we define A 1,h , A 2,h , . . . , A δ,h and put A ℓ,h ∪ B h into G for all 1 ≤ ℓ ≤ δ as follows. Let x 1 , x 2 , . . . , x d be the elements of D, and A 1,h be a (v − 4)-element set containing x 1 and e α (with α defined in the previous paragraph), and for 2 ≤ ℓ ≤ d let A ℓ,h be an arbitrary (v − 4)-element subset of C containing x 1 , x ℓ . (We need v − 4 ≥ 3 to be able to make the choice of A 1,h .) Finally, let A d+1,h , A d+2,h , . . . , A δ,h be distinct (v − 4)-element subsets of C \ {x 1 }. There are v − 3 such subsets, each missing one element of C \ {x 1 }. We take them one by one, starting with those that miss an element from D \ {x 1 }. The choice of A 1,h verifies item 4 of the F -good property and items 1 and 2 hold by definition.
To see item 3, let 1 ≤ i < j ≤ m. We need to create a copy of a Berge-F ′ . Vertices of D will play the role of N(u 1 ) ∩ N(u 2 ), A 1,i ∪ B i , A 2,i ∪ B i , . . . , A d,i ∪ B i will play the role of the edges connecting u 1 to all the vertices of D and similarly A 1,j ∪ B j , A 2,j ∪ B j , . . . , A d,j ∪ B j will play the role of the edges connecting u 2 to all the vertices of D. To finish the Berge copy of F ′ we will connect both u 1 and u 2 to all the vertices in C \ D (thus in fact we present a Berge copy of K 2,v−2 , which clearly contains F ′ ). We will use the hyperedges A i,d+1 , A i,d+2 , . . . , A i,d 1 −1 , A i,d+1 to connect u 1 to the vertices in C \ D. As they each contain ∈ H for all sets. All we need to check is whether Hall's condition holds: as for any two distinct sets, their union contains C \ D, the only problem can occur if A i,ℓ ∩ (C \ D) = A j,ℓ ′ ∩ (C \ D) = C \ D and |C \ D| = 1 or 2. But then by v − 2 ≥ 5, we have d ≥ 3 and thus all choices of A i,ℓ , A j,ℓ contain C \ D by the assumption that we picked those such subsets first that miss another element of D apart from x 1 . Suppose next d = 0. Then for any 1 ≤ h ≤ m let us fix π h , a permutation z 1 , z 2 , . . . , z v−2 of vertices of C with z 1 , z 2 being the endvertices of the edge e α . Now let A 1,h , A 2,h , . . . , A δ,h be cyclic intervals of length v − 4 of π h with e α ⊂ A 1,h . Then putting the sets of the form A ℓ,h ∪ B h to G will satisfy items 1 and 2 by definition, item 4 by the choice of A 1,h , and item 3 by a similar Hall-condition reasoning as in the case of d > 0. Now we are ready to prove the theorem. If δ = 0, then F contains an isolated edge, and we are done by Proposition 2.1. Otherwise by Claim 2.3 there exists an F -good hypergraph G with O(n) hyperedges, and by Claim 2.2 any F -saturated extension of G has a linear number of hyperedges.

Concluding remarks
For any graph F , integer r ≥ 2 and enumeration π : G 1 , G 2 , . . . , G ( n r ) of [n] r we can define a greedy algorithm that outputs a Berge-F -saturated r-uniform hypergraph G as follows: we let G 0 = ∅, and then for any i = 1, 2, . . . , n r we let G i = G i−1 ∪ {G i } if G i−1 ∪ {G i } is Berge-F -free, and G i = G i−1 otherwise. Clearly, G π,r = G ( n r ) is Berge-F -saturated.