Minimizing the number of complete bipartite graphs in a K_s-saturated graph
Beka Ergemlidze, Abhishek Methuku, Michael Tait, Craig Timmons
aa r X i v : . [ m a t h . C O ] J a n Minimizing the number of complete bipartite graphs in a K s -saturated graph Beka Ergemlidze ∗ Abhishek Methuku † Michael Tait ‡ Craig Timmons § January 5, 2021
Abstract
A graph G is F -saturated if it contains no copy of F as a subgraph but the addition ofany new edge to G creates a copy of F . We prove that for s ≥ t ≥
2, the minimumnumber of copies of K ,t in a K s -saturated graph is Θ( n t/ ). More precise results areobtained when t = 2 where the problem is related to Moore graphs with diameter 2 andgirth 5. We prove that for s ≥ t ≥
3, the minimum number of copies of K ,t in an n -vertex K s -saturated graph is at least Ω( n t/ / ) and at most O ( n t/ / ). These resultsanswer a question of Chakraborti and Loh. General estimates on the number of copies of K a,b in a K s -saturated graph are also obtained, but finding an asymptotic formula remainsopen. Let F be a graph with at least one edge. A graph G is F -free if G does not contain F as asubgraph. The study of F -free graphs is central to extremal combinatorics. Tur´an’s Theorem,widely considered to be a cornerstone result in graph theory, determines the maximum numberof edges in an n -vertex K s -free graph. An interesting class of F -free graphs are those that aremaximal with respect to the addition of edges. We say that a graph G is F -saturated if G is F -free but the addition of an edge joining any pair of nonadjacent vertices of G creates a copyof F . The function sat( n, F ) is the saturation number of F , and is defined to be the minimumnumber of edges in an n -vertex F -saturated graph. In some sense, it is dual to the Tur´anfunction ex( n, F ) which is the maximum number of edges in an n -vertex F -saturated graph.One of the first results on graph saturation is a theorem of Erd˝os, Hajnal, and Moon [10]which determines the saturation number of K s . They proved that for 2 ≤ s ≤ n , there is aunique n -vertex K s -saturated graph with the minimum number of edges. This graph is the joinof a complete graph on s − n − s + 2 vertices, denoted K s − + K n − s +2 . The Erd˝os-Hajnal-Moon Theorem was proved in the 1960s and since then, ∗ Department of Mathematics and Statistics, University of South Florida, Tampa, Florida, U.S.A. E-mail: [email protected] † School of Mathematics, University of Birmingham, United Kingdom. E-mail: [email protected] .Research is supported by the EPSRC grant number EP/S00100X/1 (A. Methuku). ‡ Department of Mathematics & Statistics, Villanova University, U.S.A. E-mail: [email protected] . Research is supported in part by National Science Foundation grantDMS-2011553. § Department of Mathematics and Statistics, California State University Sacramento, U.S.A. E-mail: [email protected] . Research is supported in part by Simons Foundation Grant n, F ) concerns the minimum number of edges in an F -saturated graph.More generally, one can ask for the minimum number of copies of H in an n -vertex F -saturatedgraph. Let us write sat( n, H, F ) for this minimum. This function was introduced in [18]and was motivated by the well-studied generalized Tur´an function whose systematic studywas initiated by Alon and Shikhelman [2]. Recalling that the Erd˝os-Hajnal-Moon Theoremdetermines sat( n, K s ) = sat( n, K , K s ), it is quite natural to study the generalized functionsat( n, K r , K s ), where 2 ≤ r < s . Answering a question of Kritschgau, Methuku, Tait andTimmons [18], Chakraborti and Loh [5] proved that for every 2 ≤ r < s , there is a constant n r,s such that for all n ≥ n r,s ,sat( n, K r , K s ) = ( n − s + 2) (cid:18) s − r − (cid:19) + (cid:18) s − r (cid:19) . Furthermore, they showed that K s − + K n − s +2 is the unique graph that minimizes the numberof copies of K r among all n -vertex K s -saturated graphs for n ≥ n r,s . They proved a similarresult for cycles where the critical point is that K s − + K n − s +2 is again the unique graph thatminimizes the number of copies of C r among all n -vertex K s -saturated graphs for n ≥ n r,s under some assumptions on r in relation to s (see Theorem 1.4 below). Chakraborti and Lohthen asked the following question (Problem 10.5 in [5]). Question 1.1
Is there a graph H for which K s − + K n − s +2 does not (uniquely) minimize thenumber of copies of H among all n -vertex K s -saturated graphs for all large enough n ? Here we answer this question positively and show that there are graphs H for which K s − + K n − s +2 is not the unique extremal graph.We begin by stating our first two results, Theorems 1.2 and 1.3, where H = K ,t . Together,they demonstrate a change in behaviour between the cases H = K , and H = K ,t with t > Theorem 1.2 (i) For n ≥ (cid:18) n (cid:19) − n / ≤ sat( n, K , , K ) ≤ (cid:18) n − (cid:19) . (ii) For n ≥ s ≥
4, sat( n, K , , K s ) = ( s − (cid:18) n − (cid:19) + ( n − s + 2) (cid:18) s − (cid:19) . Furthermore, K s − + K n − s +2 is the unique n -vertex K s -saturated with minimum number ofcopies of K , . Theorem 1.3
For integers n ≥ s ≥ and t ≥ , sat( n, K ,t , K s ) = Θ( n t/ ) . s, t ≥ n is large enough in terms of t , K s − + K n − s +2 does not minimize the number of copies of K ,t among all n -vertex K s -saturated graphs.Indeed, K s − + K n − s +2 has Θ( n t ) copies of K ,t . Interestingly, the special case of determiningsat( n, K , , K ) is related to the existence of Moore graphs. This is discussed further in theConcluding Remarks section, but whenever a Moore graph of diameter 2 and girth 5 exists,this graph will have fewer copies of K , than K + K n − = K ,n − . Thus, any potential resultthat determines sat( n, K , , K ) exactly would have to take this into account.The graph used to prove the upper bound of Theorem 1.3 is a K s -saturated graph withmaximum degree at most c s n / . This graph was constructed by Alon, Erd˝os, Holzman, andKrivelevich [1] and it is structurally very different from K s − + K n − s +2 . Using this graph onecan prove a more general upper bound that applies to any connected bipartite graph. This willbe stated in Theorem 1.5 below.Next we turn our attention to counting copies of K ,t (for t ≥
2) in K s -saturated graphs.The graph K + K n − is K -saturated and K ,t -free. Thus, sat( n, K ,t , K ) = 0 for all t ≥ t = 2 and s ≥ n, K , , K s ) = (1 + o (1)) (cid:18) s − (cid:19)(cid:18) n (cid:19) . (1)Observe that the graph K s − + K n − s +2 has (cid:18) s − (cid:19)(cid:18) n − s + 22 (cid:19) + (cid:18) s − (cid:19) ( n − s + 2) + (cid:18) s − (cid:19) copies of K , and this gives the upper bound in (1). Now the focus of [5] was on countingcomplete graphs and counting cycles, so here the above result is stated in terms of K , but ofcourse K , = C . However, it is important and relevant to this work to mention the followingtheorem of Chakraborti and Loh which shows that K s − + K n − s +2 minimizes the number ofcopies of C r in certain cases. Theorem 1.4 (Chakraborti and Loh [5])
Let s ≥ and r ≥ if r odd, and r ≥ √ s − if r is even. There is an n r,s such that for all n ≥ n r,s , the graph K s − + K n − s +2 minimizesthe number of copies of C r over all n -vertex K s -saturated graphs. Moreover, when r ≤ s − ,this is the unique extremal graph. It is conjectured in [5] that K s − + K n − s +2 is the unique graph that minimizes the numberof copies of C r among all K s -saturated graphs. Currently it is only known that K s − + K n − s +2 minimizes the number of copies of K r (Erd˝os-Hajnal-Moon for r = 2 and [5] for r > C r under certain assumptions (stated in Theorem 1.4).Theorem 1.3 shows K s − + K n − s +2 does not minimize the number of copies of K ,t . We extendthis to K a,b with 1 ≤ a + 1 < b using the following theorem. Theorem 1.5
Let F be a connected bipartite graph with parts of size a and b with ≤ a +1 < b .If s ≥ be an integer, then sat( n, F, K s ) = (cid:26) if a > s − ,O ( n ( a + b +1) ) if a ≤ s − where the implicit constant can depend on a , b , and s . K ,t must therebe in a K s -saturated graph? In this direction we prove the following. Theorem 1.6
Let s ≥ and t ≥ be integers. There is a positive constant C such that sat( n, K ,t , K s ) ≥ Cn t/ / . By Theorem 1.5, sat( n, K ,t , K s ) ≤ O s,t ( n t/ / ) for s ≥ t ≥
3, so that there is a gap inthe exponent in the upper and lower bounds.Saturation problems with restrictions on the degrees have also been well-studied. Duffusand Hanson [7] investigated triangle-saturated graphs with minimum degree 2 and 3. Day [8]resolved a 20 year old conjecture of Bollob´as [15] which asked for a lower bound on the numberof edges in K s -saturated graphs with minimum degree t . Gould and Schmitt [14] studied K t -saturated graphs (where K t is the complete t -partite graph with parts of size 2) with agiven minimum degree. Furthermore, K s -saturated graphs with restrictions on the maximumdegree were studied in [1, 13, 19]. Turning to generalized saturation numbers, as a step towardsgeneralizing Day’s Theorem, Curry et. al. [6] proved bounds on the number of triangles in a K s -saturated graph with minimum degree t . Motivated by these results we prove a lower boundon the number of copies of K a,b in K s -saturated graphs in terms of its minimum degree. Theorem 1.7
Let s ≥ and ≤ a < b be integers with a ≤ s − . If G is an n -vertex K s -saturated graph with minimum degree δ ( G ) , then G contains at least c (cid:18) n − δ ( G ) − δ ( G ) a − (cid:19) b/ copies of K a,b for some constant c = c ( s, a, b ) > . Theorem 1.7 shows that if 0 ≤ α < a − and δ ( G ) ≤ κn α for some κ >
0, then G contains atleast cn b/ − α ( a − copies of K a,b . In particular, when δ ( G ) is a constant, we obtain Ω( n t/ )copies of K ,t . This improves the lower bound of Theorem 1.6, but comes at the cost of aminimum degree assumption.In the next subsection we give the notation that will be used in our proofs. Section 2contains the proofs of Theorems 1.2 and 1.3. Section 3 contains the proofs of Theorems 1.5,1.6, and 1.7. For graphs F and G , we write N ( F, G ) for the number of copies of F in G . For a graph G and x, y ∈ V ( G ), write N ( x ) for the neighborhood of x , and N ( x, y ) for N ( x ) ∩ N ( y ). Moregenerally, if X ⊆ V ( G ) and v ∈ V ( G ), then N ( v, X ) is the set of vertices adjacent to all of thevertices in { v } ∪ X , and N ( X ) is the set of vertices adjacent to all vertices in X . We write d ( v ) = | N ( v ) | , d ( X ) = | N ( X ) | , and d ( v, X ) = | N ( v, X ) | . The set N [ v ] = N ( v ) ∪ { v } is theclosed neighborhood of v . For a graph G , let e ( G ) denote the number of edges in G .For a hypergraph H , d H ( v ) is the number of edges in H containing v . Similarly, d H ( X ) and d H ( v, X ) is the number of edges in H containing X and { v } ∪ X , respectively.4 Bounds on sat ( n, K ,t , K s ) Since the graph K s − + K n − s +2 is K s -saturated, by counting the number of copies of K , init, we have sat( n, K , , K s ) ≤ ( s − (cid:18) n − (cid:19) + ( n − s + 2) (cid:18) s − (cid:19) . (2)In particular, if s = 3 we have sat( n, K , , K ) ≤ (cid:0) n − (cid:1) . We now prove a matching lowerbound up to an error term of order O ( n / ). Let G be an n -vertex K -saturated graph. If e ( G ) ≥ √ n − n , then for n ≥ N ( K , , G ) = X v ∈ V ( G ) (cid:18) d ( v )2 (cid:19) ≥ n (cid:18) e ( G ) /n (cid:19) = 2 e ( G ) n − e ( G ) ≥ (cid:18) n (cid:19) − n / . Now assume that e ( G ) < √ n − n . If x and y are not adjacent, then since G is K -saturated, x and y must be joined by a path of length 2. Hence, N ( K , , G ) ≥ e ( G ) = (cid:18) n (cid:19) − e ( G ) ≥ (cid:18) n (cid:19) − n / . This completes the proof of (i) of Theorem 1.2. To prove (ii) of Theorem 1.2, it suffices to showthat for n ≥ s ≥ n, K , , K s ) ≥ ( s − (cid:18) n − (cid:19) + ( n − s + 2) (cid:18) s − (cid:19) , since (2) holds. Let G be an n -vertex K s -saturated graph with n ≥ s ≥
4. Kim, Kim, Kostochkaand O [17, Theorem 2.1] proved that X v ∈ V ( G ) ( d ( v ) + 1)( d ( v ) + 2 − s ) ≥ ( s − n ( n − s + 1) . (3)It is easy to check that X v ∈ V ( G ) ( d ( v ) + 1)( d ( v ) + 2 − s ) = X v ∈ V ( G ) ( d ( v ) − d ( v ) + (4 − s ) X v ∈ V ( G ) d ( v ) + (2 − s ) n. (4)Therefore, combining (3) and (4), we have X v ∈ V ( G ) ( d ( v ) − d ( v ) ≥ ( s − n ( n − s + 1) + ( s − e ( G ) + ( s − n. (5)By the Erd˝os-Hajnal-Moon Theoremsat( n, K s ) = ( s − n − s + 2) + (cid:18) s − (cid:19) , K s − + K n − s +2 is the unique n -vertex K s -saturated with sat( n, K s ) edges. Thus,2 e ( G ) ≥ s − n − s + 2) + 2 (cid:18) s − (cid:19) = ( s − n − s + 1) . Plugging this into (5) we get that if s ≥ X v ∈ V ( G ) ( d ( v ) − d ( v ) ≥ ( s − n ( n − s + 1) + ( s − s − n − s + 1) + ( s − n. Dividing through by 2 and simplifying the right-hand side yields X v ∈ V ( G ) (cid:18) d ( v )2 (cid:19) ≥ ( s − (cid:18) n − (cid:19) + ( n − s + 2) (cid:18) s − (cid:19) , where equality holds only if G = K s − + K n − s +2 . This completes the proof of Theorem 1.2. Now we prove a lower bound on the number of copies of K ,t in a K s -saturated graph that givesthe correct order of magnitude for all t ≥ Proposition 2.1
Let n ≥ s ≥ and t ≥ be integers. Then sat( n, K ,t , K s ) ≥ (cid:18) √ s − t (cid:19) t n t/ + O s,t ( n t/ ) . Proof.
Let G be an n -vertex K s -saturated graph. Kim, Kim, Kostochka and O [17, Theorem1.1] proved that X v ∈ V ( G ) d ( v ) ≥ ( n − ( s −
2) + ( s − ( n − s + 2) (6)and that equality holds if and only if G is K s − + K n − s +2 , except for in the case that s = 3where equality holds if and and only if G is K + K n − or a Moore graph. By the Power MeansInequality, X v ∈ V ( G ) d ( v ) ≤ n − /t X v ∈ V ( G ) d ( v ) t /t . (7)Combining (6) and (7) with the inequality P v ∈ V ( G ) d ( v ) t ≤ t t P v ∈ V ( G ) (cid:0) d ( v ) t (cid:1) and rearranging,we obtain that N ( K ,t , G ) is equal to X v ∈ V ( G ) (cid:18) d ( v ) t (cid:19) ≥ (( n − ( s −
2) + ( s − ( n − s + 2)) t/ t t n t/ − = (cid:18) √ s − t (cid:19) t n t/ + O s,t ( n t/ ) . This completes the proof of Proposition 2.1.
Proposition 2.2
Let s ≥ and t ≥ be integers. For sufficiently large n , sat( n, K ,t , K s ) ≤ c ts n t/ t ! where c s is a constant depending only on s . roof. By a result of Alon, Erd˝os, Holzman, and Krivelevich, for each s ≥ n , there is a K s -saturated graph G with maximum degree c s √ n (the constant c s satisfies c s → s as s → ∞ ). The number of copies of K ,t in G is then X v ∈ V ( G ) (cid:18) d ( v ) t (cid:19) ≤ n (cid:18) ∆( G ) t (cid:19) ≤ c ts n t/ t ! . Proof of Theorem 1.3.
Theorem 1.3 follows immediately from Propositions 2.1 and 2.2. ( n, K ,t , K s ) with s ≥ and t ≥ ( n, K ,t , K s ) We begin this section with a basic lemma on counting copies of a graph F in a graph G withmaximum degree ∆. It is likely that this lemma, as well as Lemma 3.2, are known. Lemma 3.1
Let F be a connected bipartite graph with parts of size a and b . If G is an n -vertexgraph with maximum degree ∆ , then N ( F, G ) ≤ n ∆ a + b − . Proof.
We will prove the lemma by counting the number of possible embeddings of F in G .Let d be the diameter of F , and x be a vertex in F . For 0 ≤ i ≤ d , let N i ( x ) be the set ofvertices at distance i from x in F . We count embeddings of F in G by starting with the vertex x , and then proceeding through N ( x ), then N ( x ) and so on. There are n ways to choose avertex in G that corresponds to x . Suppose that v x is the chosen vertex in G . The verticesin G corresponding to those in N ( x ) must be neighbors of v x in G and so there are at most∆ | N ( x ) | possibilities. This process is then repeated on N ( x ), N ( x ), and so on. The crucialpoint is that each time a vertex of F is embedded in G , it is a neighbor (in G ) of a previouslyembedded vertex (from F ). Therefore, the number of possible embeddings of F in G is at most n ∆ | N ( x ) | ∆ | N ( x ) | · · · ∆ | N d ( x ) | = n ∆ a + b − . Here we have used the assumption that since F is a connected graph with diameter d , we havethe partition { x } ∪ N ( x ) ∪ N ( x ) ∪ · · · ∪ N d ( x ) = V ( F ) . Lemma 3.2
Let F be a connected bipartite graph with parts of size a and b . For any n -vertexgraph G , N ( K a,b , G ) ≤ N ( F, G ) . Proof. If G has no K a,b , then the lemma is trivial. Suppose K is a copy of K a,b in G . Then,since F is a subgraph of K a,b , we have that F is a subgraph of K so G has a copy of F .Moreover, since any two different copies of K a,b have different vertex sets, they give rise todifferent copies of F . Thus, for each copy of K a,b in G we obtain a copy of F , and no copy of F will be obtained twice in this way. This proves Lemma 3.2.We are now ready to prove Theorem 1.5. 7 roof of Theorem 1.5. If a > s −
2, then K s − + K n − s +2 is K s -saturated with no copies of F . Indeed, a copy of F would need at least a vertices from the K s − , but a > s − a ≤ s −
2. Let G sq be the K s -saturated graph constructed in [1] where n (andthus q ) is chosen large enough so that b < q +12 . There is a constant c s > G sq ) ≤ c s √ n . By Lemma 3.1, the number of copies of F in G sq is at most nc a + b − s n (1 / a + b − = c a + b +1 s n (1 / a + b +1) .We conclude this subsection by showing that the graph G sq used in the proof of Theorem1.5 cannot be used to further improve upon the upper bound of O ( n ( a + b +1) ) when F = K a,b .Since we are showing that G sq cannot be used to improve the upper bound, we will be brief inour argument. We will use the same terminology as in [1], but one point at which we differ isthe notation we use for a vertex. A vertex in G sq is determined by its level, place, type, andcopy. A vertex at level i , place j , type t , and copy c will be written as(( i − q + j, t, c ) . First, take n large enough so that b < q +12 . Choose a sequence i , i , . . . , i a of levels with1 ≤ i < i < · · · < i a ≤ q +12 . Likewise, choose a sequence of b levels q +12 ≤ i a +1 < i a +2 < · · ·
2] which can be done in s − ≤ c , c , . . . , c a + b ≤ s − s − a + b ways.Using the definition of G sq , one finds that the a vertices in the set { (( i z − q + j , t , c z ) : 1 ≤ z ≤ a } are all adjacent to the b vertices in the set { (( i z − q + ( j + 1) q , ( t + 1) s − , c z ) : a + 1 ≤ z ≤ a + b } (here ( j + 1) q is the unique integer ζ in { , , . . . , q } for which j + 1 ≡ ζ (mod q ), and ( t + 1) s − is the unique integer ζ ′ in { , , . . . , s − } for which t + 1 ≡ ζ ′ (mod s − K a,b in G sq and so the number of K a,b in G sq is at least (cid:18) (1 / q + 1) a (cid:19)(cid:18) (1 / q + 1) b (cid:19) q ( s − s − a + b ≥ C s,a,b q a + b +1 ≥ Cn (1 / a + b +1) . By Lemmas 3.2 and 3.1, G sq is a K s -saturated n -vertex graph with Θ s,a,b ( n (1 / a + b +1) ) copiesof K a,b . ( n, K ,t , K s ) First we prove Theorem 1.6.
Proof of Theorem 1.6.
Let G be a K s -saturated graph on n vertices. Note that we canassume e ( G ) ≤ n . (8)Otherwise, a theorem of Erd˝os and Simonovits [9] implies that there is a positive constant γ such that N ( K ,t , G ) ≥ γ e ( G ) t n t − = Ω( n t +2 ) , (9)8roving Theorem 1.6.Let K − be the graph consisting of 4 vertices and 5 edges obtained by removing an edgefrom K . For a copy of K − with vertices x, y, u, v , where uv / ∈ E ( G ), let xy be called the baseedge of this K − . We estimate the number of copies of K − in a K s -saturated graph G .For every u, v with uv / ∈ E ( G ) there is a set S such that S ⊆ N ( u, v ) and S induces a K s − in G . Therefore, there are at least (cid:0) s − (cid:1) pairs x, y ∈ S such that u, v, x, y form a copy of K − .On the other hand, every xy ∈ E ( G ) is the base edge of at most (cid:0) d ( x,y )2 (cid:1) copies of K − in G .Therefore, X xy ∈ E ( G ) (cid:18) d ( x, y )2 (cid:19) ≥ N ( K − , G ) ≥ X uv ∈ E ( G ) (cid:18) s − (cid:19) ≥ e ( G ) (8) ≥ n . Thus, there is a constant c t = c ( t ) such that the following holds: N ( K ,t , G ) ≥ X xy ∈ E ( G ) (cid:18) d ( x, y ) t (cid:19) ≥ t t X xy ∈ E ( G ) (cid:18) d ( x, y )2 (cid:19) t − t t ! ≥ e ( G ) t t P xy ∈ E ( G ) (cid:0) d ( x,y )2 (cid:1) e ( G ) ! t − e ( G ) ≥ ( n / t t t e ( G ) t − − e ( G ) = ( n / t − t t e ( G ) t/ t t e ( G ) t − ≥ c t n t e ( G ) t − . Combining this with (9) we get N ( K ,t , G ) ≥ min { γ e ( G ) t n t − , c t n t e ( G ) t − } . Let e ( G ) = n α , then N ( K ,t , G ) ≥ min n γn αt − t +2 , c t n t − αt/ α o . Choosing α = t − t − and C = min { γ, c t } , we get the desired lower bound Cn t − t − + >Cn t + .Next we turn to the proof of Theorem 1.7. We need the following lemma. Lemma 3.3
Let s ≥ and ≤ a ≤ b be integers with a ≤ s − . Suppose that G is an n -vertex K s -saturated graph with vertex set V . There is a constant c = c ( s, a, b ) such that for any v ∈ V ,there are at least c (cid:18) n − d ( v ) − d ( v ) a − (cid:19) b/ copies of K a,b containing v . Proof.
Let v ∈ V . For each u ∈ V \ N [ v ], there is a set S u ⊂ N ( v ) such that S u induces a K s − in G . Fix such an S u and define an ( s − H to have vertex set V \{ v } ,and edge set E ( H ) = {{ u } ∪ S u : u ∈ V \ N [ v ] } . By construction, H has n − d ( v ) − V \ N [ v ] and s − N ( v ). Also, no twoedges of H contain the same vertex from V \ N [ v ]. In what follows, we will add the subscript H if we are referring to degrees in H , and no subscript will be included if we are referring todegrees or neighborhoods in G . 9y averaging, there is a set X ∈ (cid:0) N ( v ) a − (cid:1) such that d H ( X ) ≥ (cid:0) s − a − (cid:1) ( n − d ( v ) − (cid:0) d ( v ) a − (cid:1) . We then have X y ∈ N ( v,X ) d H ( y, X ) ≥ d H ( X )( s − − | X | ≥ c n − d ( v ) − d ( v ) a − (10)for some constant c = c ( s, a ) >
0. The number of K a,b with X ∪ { y } forming the part of size a ( y is an arbitrary vertex from N ( v, X )) and v in the part of size b is at least X y ∈ N ( v,X ) (cid:18) d H ( y, X ) b − (cid:19) ≥ d ( v, X ) (cid:18) c ( n − d ( v ) − d ( v,X ) d ( v ) a − b − (cid:19) ≥ c ( n − d ( v ) − b − d ( v, X ) b − d ( v ) ( a − b − . Here we have used convexity, (10), and c = c ( s, a, b ) is some positive constant.Recalling that | X | = a −
1, there are (cid:0) d ( v,X ) b (cid:1) copies of K a,b where { v } ∪ X is the part of size a and the part of size b is contained in N ( v ) \ X . Thus, for some constant c = c ( s, a, b ) > K a,b that contain v is at least c ( n − d ( v ) − b − d ( v, X ) b − d ( v ) ( a − b − + c d ( v, X ) b . By considering cases as to which is this the bigger term in this sum, we find that in both cases,there are at least c (cid:18) n − d ( v ) − d ( v ) a − (cid:19) b/ copies of K a,b containing v .Applying Lemma 3.3 to a vertex v with d ( v ) = δ ( G ) proves Theorem 1.7. An interesting open problem is determining the minimum number of copies of K , in a K -saturated graph. There is a connection between this problem and Moore graphs with diameter2 and girth 5. It is easy to check that an n -vertex Moore graph with diameter 2 and girth 5is K -saturated, and it is regular with degree d = √ n − n (cid:0) d (cid:1) = n (cid:0) √ n − (cid:1) copies of K , , and for all n ≥
3, this value is less than (cid:0) n − (cid:1) which is the number of copies of K , in K + K n − = K ,n − . Furthermore, one can duplicate vertices of a Moore graph andpreserve the K -saturated property (where each duplicated vertex has the same neighborhoodas the original vertex). Duplicating a vertex of the Petersen graph will lead to an 11-vertex K -saturated graph with 42 copies of K , , but K , has 45 copies of K , . Starting from theHoffman-Singleton graph, one can duplicate a vertex up to 4 times and we can still have fewercopies of K , compared to the number of copies of K , in K ,n − . Duplicating a single vertexis not necessarily the optimal way to minimize the number of copies of K , , but the point isthat there are other graphs besides the Moore graphs that have fewer copies of K , than thenumber of copies of K , in K ,n − . 10t would also be interesting to determine the order of magnitude of sat( n, K ,t , K s ). Thereis a gap in the exponents (which is discussed in the introduction) and it would be nice to closethis gap. It is not clear if our lower or upper bound is closer to the correct answer.Another potential approach to studying sat( n, H, F ) is via the random F -free process. Thisrandom process orders the pairs of vertices uniformly and then adds them one by one subjectto the condition that adding an edge does not create a copy of F . The resulting graph is then F -saturated. This process was first considered in [4, 11, 20, 22] and has since been studiedextensively. If X H,F is the random variable that counts the number of copies of H in theoutput of this process, then we have that sat( n, H, F ) ≤ E ( X H,F ). It would be interesting todetermine for which graphs H and F that this approach gives better bounds than the explicitconstructions that are currently known. References [1] N. Alon, P. Erd˝os, R. Holzman, M. Krivelevich, On k -saturated graphs with restrictionson the degrees, J. Graph Theory
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