aa r X i v : . [ m a t h . C O ] J a n A construction for bipartite Tur´an numbers
Ivan Livinsky
Abstract
We consider in detail the well-known family of graphs G ( q, t ) that establish an asymptoticlower bound for Tur´an numbers ex( n, K ,t +1 ). We prove that G ( q, t ) for some specific q and t also gives an asymptotic bound for K , and for some higher complete bipartite graphsas well. The asymptotic bounds we prove are the same as provided by the well-knownNorm-graphs. In his 1996 paper [1] for a prime power q and t | q − G ( q, t ) definedas follows. Let F = GF( q ) be a finite field having q elements. Let H ⊆ F ∗ be a subgroup of themultiplicative group containing t elements. Define V ( G ( q, t )) = ( F × F \ { (0 , } ) / ∼ where two pairs ( a , b ), ( a , b ) are equivalent iff there exists h ∈ H such that a = ha and b = hb . Hence, G ( q, t ) has ( q − /t vertices. We write h a, b i for the equivalence classcontaining pair ( a, b ). Two vertices h a, b i and h x, y i are joined by an edge iff ax + by ∈ H. We can easily see that in this way a simple graph is correctly defined. For a pair ( a, b ) and h ∈ H fixed the equation ax + by = h defines a line containing q points in F and any two points on this line are pairwise non-equivalent. Therefore, degree of each vertex is either q or q −
1. Thus, G ( q, t ) contains at least t ( q − q −
1) edges. 1 heorem 1.1 ([1]) . For arbitrary prime power q and t | q − the graph G ( q, t ) is K ,t +1 -free. We reproduce F¨uredi’s proof here since we will use some of its steps later.
Proof.
If a vertex h x, y i is attached to two distinct vertices h a , b i and h a , b i then there exist h , h ∈ H such that the following system of linear equations a x + b y = h ,a x + b y = h holds.First, we show that the matrix of the system (cid:18) a b a b (cid:19) has full rank. Indeed, both lines are nonzero and if a = ca and b = cb for some c ∈ F ∗ thenwe have also h = ch implying that c = h h − ∈ H . Thus, we have that h a , b i = h a , b i andwe arrive to a contradiction.Therefore, for arbitrary h and h there exists a unique solution ( x, y ). We have t choicesfor h and h totally, and all the solutions are divided into t classes. Therefore, there are atmost t vertices attached to both h a , b i and h a , b i .We show that for some specific choice of q and t the graphs G ( q, t ) in a similar way givelower bounds for larger bipartite graphs as well. In fact, using them we can obtain the sameasymptotic bounds as given by the well-known Norm-graphs [4]. K , Let q be a prime power. Consider the graph G = G ( q , q + 1). Let F = GF( q ). Theorem 2.1.
For arbitrary prime power q the graph G ( q , q + 1) is K , -free. We will need an auxiliary lemma.
Lemma 2.2.
Let H ⊂ F ∗ , | H | = q + 1 , be a subgroup. Let a, b ∈ F , a = 0 , b = 0 . Then theequation ax + by = 1 has at most two solutions ( x, y ) for x, y ∈ H . roof. Assume that ( x, y ) is a solution. Then by = 1 − ax , x q +1 = y q +1 = 1 and b q +1 = ( by ) q +1 = (1 − ax ) q +1 = (1 − ax ) q (1 − ax )= (1 − a q x q )(1 − ax ) = (1 − a q x )(1 − ax ) = 1 − ax − a q x + a q +1 . Therefore, x is a solution to a proper quadratic equation ax − ( a q +1 − b q +1 + 1) x + a q = 0 , and y = − axb .Therefore, there are at most two possible pairs ( x, y ). Proof of Theorem 2.1.
Consider three distinct vertices h a , b i , h a , b i , h a , b i and assume thatanother vertex h x, y i is attached to all of them. We have a system of equations a x + b y = h ,a x + b y = h ,a x + b y = h . We know from the proof of Theorem 1.1 that the matrix (cid:18) a b a b (cid:19) has rank two. Therefore, the third equation must be a linear combination of the first two. Thatis, there must exist uniquely defined coefficients α, β ∈ F such that αa + βa = a ,αb + βb = b ,αh + βh = h . Moreover, α = 0 and β = 0 since otherwise we will have equality between the initial vertices.Consider the last equation. Let r = h h − and s = h h − . Then αr + βs = 1 . However, according to Lemma 2.2 there are at most two solutions ( r, s ) to this equation. There-fore, there are at most 2( q + 1) triples ( h , h , h ) such that the original system has a solution.These triples define at most two vertices h x, y i . Therefore, G is K , -free.We have that | V ( G ) | = q − q +1 = ( q + 1)( q −
1) = q − q + q −
1. We can also give an exactformula for the number of edges. 3 heorem 2.3.
Let G = G ( q , q + 1) . If q is odd then | E ( G ) | = ( q − q + q − q + 1) . If q = 2 k then | E ( G ) | = ( q − q + q − q ) .Proof. If q is odd then q ≡ − F . Therefore, the equation x + y = c has exactly | F | − q − c = 0. If q = 2 k then this equation defines a line in F and has | F | = q solutions.A vertex h x, y i has degree q − G iff x + y ∈ H according to the construction. Therefore,the number of vertices of degree q − q − q odd, and to q for q even. Thus,for q odd | E ( G ) | = 12 (cid:0) ( q − q + q ) q + ( q − (cid:1) = 12 ( q − q + q − q + 1) . And for q even | E ( G ) | = 12 (cid:0) ( q − q + q − q + q ( q − (cid:1) = 12 ( q − q + q − q ) . Therefore, for n = q − q + q − K , -free graph with n + n + O( n ) edges.Together with the upper bound [2] for K , this gives the asymptotic formulaex( n, K , ) = 12 n (1 + o (1)) . We can also obtain a lower bound for the graphs K , t +1 . Theorem 2.4.
For arbitrary prime power q and t | q − the graph G ( q , t ( q +1)) is K , t +1 -free.Proof. We follow the same steps as in the proof of Theorem 2.1. We arrive to the equation αr + βs = 1 , where now we assume that r, s belong to the subgroup H of order t ( q + 1). Let H ′ be a subgroupof order q + 1. Then we can choose t coset representatives g , . . . , g t of H/H ′ . We have that r = g i r ′ , s = g j s ′ for some r ′ , s ′ ∈ H ′ and α ′ r ′ + β ′ s ′ = 1 , where α ′ = g i α , β ′ = g j β . According to Lemma 2.2 this equation has at most two solutions.Since the choice of g i , g j can be arbitrary we have that there are at most 2 t pairs ( r, s ). There-fore, G is K , t +1 -free. 4his gives an asymptotic bound of the formex( n, K , t +1 ) ≥ t n (1 + o (1)) . This asymptotic bound was also proved by Mont´agh in his PhD thesis [5] using a factorizationof the Brown graph [6].
Alon, Koll´ar, R´onyai, and Szab´o introduced Norm-graphs in the papers [3, 4]. They constructeda family of graphs that were K r, ( r − -free and had n vertices and n − r (1+ o (1)) edges. Theirconstruction depended heavily on the following algebro-geometric lemma Lemma 3.1 ([3]) . Let q be a prime power, let F = GF( q r ) , and let N : F → GF( q ) , x x q + ··· + q r − be the norm map of F over GF( q ) . Let c , . . . , c r , d , . . . , d r ∈ F be some elements. If d i = d j for i = j then the system of equations N ( x + d ) = c ,N ( x + d ) = c ,. . .N ( x + d r ) = c r , has at most r ! solutions for x ∈ F . Using Lemma 3.1 we establish the same asymptotic bound using graphs G ( q, t ). Theorem 3.2.
For arbitrary prime power q the graph G ( q r − , q r − + · · · + q + 1) is K r, ( r − -free.Proof. Let F = GF( q r − ). Let H ⊂ F ∗ be a subgroup of order q r − + · · · + q + 1. Considerdistinct r vertices h a , b i , . . . , h a r , b r i and assume that another vertex h x, y i is attached to allof them. We have another system of linear equations a x + b y = h ,a x + b y = h ,. . .a r x + b r y = h r .
5s before, we have that all equations from third to last are linear combinations of the first two.That is, for every j = 3 , . . . , r there exist uniquely defined nonzero elements α j , β j ∈ F suchthat α j a + β j a = a j ,α j b + β j b = b j ,α j h + β j h = h j . Therefore, we have a system of r − α h + β h = h ,. . .α r h + β r h = h r . Note that N ( h ) = 1 for all h ∈ H . Therefore for each j = 3 , . . . , r we have that N (cid:18) α j β j + h h (cid:19) = N (cid:18) h j h β j (cid:19) = N ( β j ) − . Moreover, we also have that N (cid:18) h h (cid:19) = 1 . Note that α j β j = 0, and α i β i = α j β j when i = j since otherwise we would get h a i , b i i = h a j , b j i .We have a system of r − r − h /h . Each solution defines a unique vertex h x, y i attached to all h a i , b i i . Therefore, G is K r, ( r − -free.We have that G = G ( q r − , q r − + · · · + q + 1) has ( q r − + 1)( q −
1) vertices and at least ( q r − − q −
1) edges. Therefore, it achieves the asymptotic lower bound of the formex( n, K r, ( r − ) ≥ n − r (1 + o (1)) . Finally, we can prove a lower bound for the graphs K r,t r − ( r − . Theorem 3.3.
For arbitrary prime power q and t | q − the graph G ( q r − , t ( q r − + · · · + q + 1)) is K r,t r − ( r − -free.Proof. Let H and H ′ be subgroups of F ∗ orders t ( q r − + · · · + q + 1) and q r − + · · · + q + 1respectively. Choose t coset representatives g , . . . , g t of H/H ′ .6e follow the same steps as in Theorem 3.2. The only difference is that in the final systemwe obtain equations of the form N (cid:18) α j β j + h h (cid:19) = N (cid:18) h j h (cid:19) N ( β j ) − , but N (cid:16) h j h (cid:17) for j = 2 , . . . , r can be any of the t elements N ( g ) , . . . , N ( g t ) only. Therefore,in this case there are at most t r − ( r − h /h . Again, each of these solutionsuniquely defines a vertex h x, y i attached to all h a i , b i i .We obtain an asymptotic boundex( n, K r,t r − ( r − ) ≥ t r − r n − r (1 + o (1)) . Therefore, our construction achieves the same asymptotic lower bounds for bipartite Tur´annumbers as the Norm-graphs do.
References [1] Z. F¨uredi,
New asymptotics for bipartite Tur´an numbers , J. Combin. Theory Ser. A 75 (1996),no. 1, 141–144.[2] Z. F¨uredi,
An upper bound on Zarankiewicz problem , Combin. Probab. Comput. 5 (1996),no. 1, 29–33.[3] J. Koll´ar, L. R´onyai, T. Szab´o,
Norm-graphs and bipartite Tur´an numbers . Combinatorica16, 399–406, 1996.[4] N. Alon, L. R´onyai, and T. Szab´o,
Norm-graphs: variations and applications
J. Combin.Theory Ser. B, 76(2), 280–290, 1999.[5] B. Mont´agh,
Unavoidable substructures , PhD Thesis, University of Memphis, May 2005.[6] W. G. Brown,