GRAPHS WITHOUT THETA SUBGRAPHS. 1. Introduction

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1 GRAPHS WITHOUT THETA SUBGRAPHS J. VERSTRAETE AND J. WILLIFORD Abstract. In this paper we give a lower bound on the greatest number of edges of any n vertex graph that contains no three distinct paths of length four with the same endpoints. The construction is algebraic in nature, arising from equations over finite fields, and is perhaps some evidence that ex(n, C 8) = Θ(n 5/4 ). 1. Introduction For a family F of finite graphs, the Turán Numbers of F are the quantities ex(n, F) denoting the maximum number of edges in an n-vertex graph that has no member of F as a subgraph. While the asymptotic value of ex(n, F) is known when F does not contain any bipartite graph, via the celebrated Erdős-Stone-Simonovits Theorem [6], the case when F contains a bipartite graph is generally more difficult. In this paper we concentrate on a particular instance of this problem for bipartite graphs, motivated by the notoriously difficut extremal problem for even cycles. Let C l = {C 3, C 4,..., C l } denote the set of cycles of length at most k. For each l 2, Erdős and Simonovits [5] conjectured ex(n, C 2l ) = Θ(n 1+1/l ) and this conjecture remains open except in the cases l {3, 4, 5, 6, 7, 10, 11}, essentially due to the existence of certain generalized polygons see [11]. A general upper bound for ex(n, C 2l ) was given by Bondy and Simonovits [1], and is also attributed to Erdős: Theorem 1. For each l 2 there exists c l such that ex(n, C 2l ) c l n 1+ 1 l. Bukh and Jiang [2] recently showed one can take c l = 80 l log l. The best lower bounds are due to Lazebnik, Ustimenko and Woldar [12] (see [11] for the discussion of the cases l {2, 3, 5}, and see [14] for a survey of extremal problems for cycles). The smallest open case, the cycle of length eight, is of particular interest. The current best lower bound is ex(n, C 8 ) = Ω(n 6/5 ) using generalized hexagons see [8, 11]. Research supported by NSF Grant DMS Research supported by NSF Grant DMS

2 2 GRAPHS WITHOUT THETA SUBGRAPHS A theta-graph, denoted θ k,l, consists of k 2 internally disjoint paths of length l with the same endpoints. A θ 2,l is simply a cycle C 2l, so the problem of determining ex(n, θ k,l ) generalizes the problem of determining ex(n, C 2l ). Faudree and Simonovits [7] showed that a θ k,l -free graph on n vertices has O k,l (n 1+1/l ) edges, which is of the same order of magnitude as the upper bound for ex(n, C 2l ) in Theorem 1. Since ex(n, C 2l ) = Θ(n 1+1/l ) when l {2, 3, 5}, we also have ex(n, θ k,l ) = Θ(n 1+1/l ) for l {2, 3, 5} and all k 2. An algebraic construction based on random low-degree polynomials over finite fields due to Conlon [4], combined with the afore-mentioned result of Faudree and Simonovits [7], gives the following theorem: Theorem 2. For each l 2, there exists k l such that for all k k l, ex(n, θ k,l ) = Θ(n 1+1/l ). The best possible value of k l from [4] is k l = exp(o(l 2 log l)). Our main theorem is as follows: Theorem 3. ex(n, θ 3,4 ) = Θ(n 5/4 ). In some sense, this gives evidence that ex(n, C 8 ) = Θ(n 5/4 ). 2. Construction of a theta-free graph Let q be an odd prime power. For convenience, throughout the analysis we let v i denote the ith co-ordinate of a vector v F 4 q. The graph G q is defined with vertex set V = F 4 q where v V is joined to w V if w 2 = v 2 + v 1 w 1 w 3 = v 4 + v 1 w 2 1 w 4 = v 3 + v 2 1w 1. By fixing a vertex w, it is not hard to show that the choice of v 1 determines a unique neighbor v of w, except in the case where v = w. Therefore each vertex has degree q or q 1. A simple computation shows that the graph has q 2 vertices of degree q 1, implying that G q has 1 2 (q5 q 2 ) edges. It is not hard to see that G q has no cycles of length four or six. Indeed, this graph is a polarity graph of the graph D(4, q), see [12]. Theorem 3 is a consequence of the following: Theorem 4. The graph G q is θ 3,4 -free.

3 GRAPHS WITHOUT THETA SUBGRAPHS 3 3. Preliminaries We begin with the following simple fact: Fact. If v, w F 4 q are distinct and have a common neighbor, then v 1 w 1. We begin by classifying the octagons in G q. Suppose there is an octagon formed by the vertices (a, l, b, m, c, n, d, o). Then we have a 2 + l 2 = a 1 l 1, l 2 + b 2 = l 1 b 1, b 2 + m 2 = b 1 m 1,... o 2 + a 2 = o 1 a 1, and alternately adding and subtracting these equations we obtain (a 1 b 1 )l 1 + (b 1 c 1 )m 1 + (c 1 d 1 )n 1 + (d 1 a 1 )o 1 = 0. Similarly, we have a 3 + l 4 = a 1 l 2 1, l 4 + b 3 = l 2 1 b 1, b 3 + m 4 = b 1 m 2 1,... o 4 + a 3 = o 2 1 a 1, and alternately adding and subtracting these equations we obtain (a 1 b 1 )l (b 1 c 1 )m (c 1 d 1 )n (d 1 a 1 )o 2 1 = 0. This gives us the three equations: (a 1 b 1 )l 1 + (b 1 c 1 )m 1 + (c 1 d 1 )n 1 + (d 1 a 1 )o 1 = 0 (1) (a 1 b 1 )l (b 1 c 1 )m (c 1 d 1 )n (d 1 a 1 )o 2 1 = 0 (2) (l 1 m 1 )a (m 1 n 1 )b (n 1 o 1 )c (o 1 l 1 )d 2 1 = 0. (3) It is convenient to let x 1 = a 1 b 1, x 2 = b 1 c 1, x 3 = c 1 d 1 and x 4 = d 1 a 1. Likewise let y 1 = l 1 m 1, y 2 = m 1 n 1, y 3 = n 1 o 1, y 4 = o 1 l 1. We note from Fact 1 that none of the x i and y i are zero, otherwise the octagon is degenerate. We also note that the cyclic change of variables a 1 l 1 b 1 m 1 c 1 n 1 d 1 o 1 a 1 preserves this system of equations, and so the changes of variables x 1 y 1 x 2 y 2 x 3 y 3 x 4 y 4 x 1 is also valid. Claim 1. Proof. From (1) we can deduce x 2 1 = x 2 3 (4) x 2 2 = x 2 4 (5) y 2 1 = y 2 3 (6) y 2 2 = y 2 4. (7) ((a 1 b 1 )l 1 + (b 1 c 1 )m 1 ) 2 ((c 1 d 1 )n 1 + (d 1 a 1 )o 1 ) 2 = 0. (8) We then multiply (2) by a 1 c 1 and subtract the above equation to get (a 1 b 1 )(b 1 c 1 )(l 1 m 1 ) 2 = (c 1 d 1 )(d 1 a 1 )(n 1 o 1 ) 2. (9)

4 4 GRAPHS WITHOUT THETA SUBGRAPHS Similarly combining (3) and (1) we get (l 1 m 1 )(m 1 n 1 )(b 1 c 1 ) 2 = (n 1 o 1 )(o 1 l 1 )(d 1 a 1 ) 2. (10) We also get (b 1 c 1 )(c 1 d 1 )(m 1 n 1 ) 2 = (d 1 a 1 )(a 1 b 1 )(o 1 l 1 ) 2 (11) (m 1 n 1 )(n 1 o 1 )(c 1 d 1 ) 2 = (o 1 l 1 )(l 1 m 1 )(a 1 b 1 ) 2 (12) Then we have from (8) (12) that x 1 x 2 y1 2 = x 3 x 4 y3 2 (13) x 2 x 3 y2 2 = x 4 x 1 y4 2 (14) y 1 y 2 x 2 2 = y 3 y 4 x 2 4 (15) y 2 y 3 x 2 3 = y 4 y 1 x 2 1. (16) By (14), x 2 x 3 x 4 x 1 = y2 4 y 2 2 (17) therefore x 2 2 x2 3 x 2 4 x2 1 = y4 4 y2 4. (18) Since none of the x i and none of the y i are zero, using (15) and (16) we obtain x 2 2 x2 3 x 2 4 x2 1 = y2 4 y2 2. (19) This implies y2 2 = y2 4. Applying cyclic changes of variables yields y2 1 = y2 3, x 2 1 = x2 3 and x2 2 = x2 4, as required. Claim 2. a 1 + c 1 = b 1 + d 1 and l 1 + n 1 = m 1 + o 1. Proof. Suppose x 1 = x 3. By (12), x 1 x 2 = x 3 x 4, so we must have x 2 = x 4. Since x 1 + x 2 + x 3 + x 4 = 0, we have x 1 + x 2 = 0, so a 1 = c 1 and b 1 = d 1. Substituting this into (1) and (2), we have (a 1 b 1 )(l 1 m 1 + n 1 o 1 ) = 0 and (a 1 b 1 )(l 2 1 m2 1 + n2 1 o2 1 ) = 0. Since a 1 b 1, l 1 m 1 + n 1 = o 1, and substituting this into l 2 1 m2 1 + n2 1 o2 1 = 0 gives 2(l 1 m 1 )(m 1 n 1 ) = 0, a contradiction. Therefore x 1 = x 3. A similar argument shows y 1 y 3, so y 1 = y 3, and the theorem follows.

5 GRAPHS WITHOUT THETA SUBGRAPHS 5 4. Proof of Theorem 3 Suppose for a contradiction that G q contains a θ 3,4, say consisting of paths (a, l, b, m, c) (a, o, d, n, c) (a, p, f, r, c). By Claim 3, we have a 1 + c 1 = b 1 + f 1 = b 1 + d 1 = d 1 + f 1 so b 1 = d 1 = f 1 and a 1 + c 1 = 2b 1. The first equation is (a 1 b 1 )l 1 + (a 1 b 1 )m 1 + (b 1 a 1 )n 1 + (b 1 a 1 )o 1 = 0 and therefore a 1 = b 1 or l 1 + m 1 = n 1 + o 1. Since l is a common neighbor of a and b, we cannot have a = b by Fact 1. If l 1 + m 1 = n 1 + o 1, then using l 1 + n 1 = m 1 + o 1 we get m 1 = n 1, which contradicts Fact 1, since m and n have c as a common neighbor. We conclude that the graph G is θ 3,4 -free. We note that for q even, this graph contains the subgraph θ 3,4. In fact, it contains θ q 2,4. This can be seen by considering the following path for any t F q with t 0, 1: (0, 0, 0, 0), (t, 0, 0, 0), (1, t, t 2, t), (t + 1, 1, 1, 1), (0, 1, 1, 1). 5. Thetas in Wenger Graphs Let q be a prime power. The Wenger graphs W n,q have vertex set equal to V 0 V 1 where V 0, V 1 are two copies of the vector space F n q. For convenience we will denote elements of V 0 with parenthesis and elements of V 1 with square brackets. The former will be called points and the latter lines. A point p is adjacent to a line l provided that the following equations are satisfied: p 2 + l 2 = p 1 l 1 p 3 + l 3 = p 2 1l 1... p n + l n = p n 1 1 l 1 Note that this definition differs from the original one in Wenger[17], but gives an isomorphic graph (see Viglione [16]). The Wenger graphs W 2,q, W 3,q, W 5,q give the correct magnitude of ex(n, C 4 ), ex(n, C 6 ), and ex(n, C 10 ), respectively, and Wenger graphs are also known to have good expansion properties (see Wenger [17] and Cioabă, Lazebnik and Li [3]). It would seem reasonable to expect that they may give good constructions for θ k,l -free graphs in general, however we show this to be false.

6 6 GRAPHS WITHOUT THETA SUBGRAPHS The following theorem generalizes the results on the existence of cycles in Wenger graphs in He, Shan and Shao [9] and Lazebnik, Thomason and Wang [10] to theta graphs. Theorem 5. For n l 6, q > l 2, the graph W n,q contains a θ q l 2,l. We note that any walk in the graph W n,q may be specified by a vertex followed by a sequence of elements of F q, which represent the first coordinates of the remaining vertices of the walk. We call a walk w 1, w 2,... w t = w 1 recurrent if w i = w i+2 for some i or w t 1 = w 2. We will be aided by the following theorem which describes when non-recurrent walks are closed: Theorem 6. Let x be a point in W n,q with first coordinate a 1 and let a 2,..., a 2t+1 = a 1, 2t n, be a sequence of elements of F q with a i a i+2 for all i and a 2t a 2. For each a i with i > 1 odd, define φ(a i ) = a i+i a i 1, and define φ(a 1 ) = a 2 a 2t. The sequence x, a 2, a 3,..., a 2t, a 2t+1 = a 1 represents a non-recurrent closed walk in W n,q if and only if every element of the subsequence a 1, a 3,..., a 2t 1 repeats, and for each odd i 2t 1 the following holds: φ(a j ) = 0. j odd a j =a i Proof. Suppose the given walk is closed in W n,q. Let x (i) denote the ith vertex on the walk (with first coordinate a i ). This is equivalent to the following equations e i,j are satisfied for each 1 i 2t, and each 1 j n 1 : e i,j := { x (i) j x (i) j + x (i+1) j = a j i a i+1 i odd + x (i+1) j = a i a j i+1 i even Taking the sums 2t ( 1) i e i,j for each j we obtain: i=1 2t 1 i odd i=1 a j i (φ(a i)) = 0 The equations reduce to A v where A is a rectangular matrix of the form:

7 GRAPHS WITHOUT THETA SUBGRAPHS a 1 a 3 a 5... a 2t 1 a 2 1 a 2 3 a a 2 2t a n 1 1 a3 n 1 a5 n 1... a2t 1 n 1 and v = (φ(a 1 ), φ(a 3 ), φ(a 5 ),..., φ(a 2t 1 )) is a vector in the null space of U. Note that v is nowhere zero by the non-recurrence of the walk. The first 2t rows of A give a Vandermonde matrix, which has a nowhere zero vector in its null space if and only if each entry in its second row repeats and the corresponding entries of v sum to zero. Conversely, suppose that every element of the subsequence a 1, a 3,..., a 2t 1 repeats, and for each odd i 2t 1 the following holds: φ(a j ) = 0. j odd a j =a i Then the equation A v = 0 holds. Taking the point with first coordinate a 1 to be the zero vector, we can recover all the coordinates of each vertex of the walk. A simple induction shows that for even i and 2 j n we have x (i) j = i 1 k odd k=1 x j k φ(x k). This implies that the line x (2t) = [a 2t, 0, 0, 0,... 0], which is adjacent to x (1), so the walk is closed. This completes the proof.. Now we complete the proof of Theorem 6. Proof. For each l {2, 3, 4, 5} we describe walks in W n,q, each starting at the point which is the zero vector, by a sequence of first coordinates of the vertices. For l = 4t, t 2, and a fixed sequence of distinct nonzero elements of a 1,..., a 2t 1, we define for each y F q the following walk: ω y : 0, y, a 1, y + 1, a 2, y,..., y, a 2t 1, y + 1, 0. By Theorem 6, this walk cannot contain a cycle, and it is non-recurrent so it must be a path. Let ω y 1 denote the sequence of first coordinates of ω y in reverse order. For y z the walk ω y ω z 1 satisfies the conditions of theorem 6, and so must be closed. In particular, that implies that ω y and

8 8 GRAPHS WITHOUT THETA SUBGRAPHS ω z, when began at the point 0, must end at the same vertex. Furthermore, these paths must be internally disjoint, since otherwise there would be a cycle containing the point 0 that does not satisfy Theorem 6, since the first coordinate 0 would not repeat. This gives a θ q,4t for q 2t. For the remaining cases, the proof is similar. For l = 4t + 2, t 2, and a fixed sequence of distinct nonzero elements of a 1,..., a 2t+1 and fixed µ 0, 1, we define for each y F q the following walk: ω y : 0, y, a 1, y + 1, a 2, y,..., y + 1, a 2t, y + µ, 0. This gives a θ q,4t+2 for q 2t + 1. For l = 4t + 1, t 2, and a fixed sequence of distinct nonzero elements of a 1,..., a 2t 3, we define for each y F q the following walk: ω x,y : 0, y, x, y + 1, 0, 1, a 1, 0, a 2, 1,..., 1, a 2t 3, 0, x, 1. Choosing paths ω x,y so that all values of x and y are distinct, x 0, a 1,... a 2t 3 and y 0, 1, 1, we get a θ min{q 3,q 2t+2},4t+1 for q 2t + 1. For l = 4t + 3, t 1, and a fixed sequence of distinct nonzero elements of a 1,..., a 2t 2, we define for each y F q the following walk: ω x,y : 0, y, x, y + 1, 0, 1, a 1, 0, a 2, 1,..., 0, a 2t 2, 1, x, 0. Choosing paths ω x,y so that all values of x and y are distinct, x 0, a 1,... a 2t 2 and y 0, 1, 1, we get a θ min{q 2t+1,q 3},4t+3 for q 2t + 1. This completes the proof of the theorem.. 6. Concluding Remarks Mellinger and Mubayi [13] constructed certain point-line incidence graphs H(7, q) based on projective geometries, and claimed that these graphs are θ 3,7 -free. However we show below that W (7, q) H(7, q), and therefore by Theorem 6, their graphs H(7, q) contains θ q 4,7 when q > 4. First we describe the graphs H(7, q), and then we show W (7, q) H(7, q). Let Σ = P G(7, q), and Σ 0 be the hyperplane (1, 0, 0, 0, 0, 0, 0, 0) (using the usual scalar product). Let A = {(0, 1, t, t 2, t 3, t 4, t 5, t 6 ) : t F q } {(0, 0, 0, 0, 0, 0, 0, 1)} be the standard rational normal curve in Σ 0. Let P be the set of points in Σ\Σ 0, and L be the set of lines of Σ that are not contained in Σ 0 and which meet A in a point. The graph H(7, q) is defined to be the point-line incidence graph of P L. Next we show W (7, q) H(7, q) by defining an explicit embedding φ of W (7, q) into H(7, q). The map φ will map points of W (7, q)

9 GRAPHS WITHOUT THETA SUBGRAPHS 9 to lines of H(7, q) and lines of W (7, q) to points of H(7, q). For a point p = (p 1, p 2,..., p 7 ) of W (7, q) we define φ(p) to be the line in L through the points (0, 1, p 1, p 2 1, p3 1,..., p6 1 ) and (1, 0, p 2, p 3, p 4,..., p 7 ). For a line l = [l 1, l 2, l 3,..., l 7 ] we define φ(l) = (1, l 1, l 2,..., l 7 ) P. This map is easily seen to be injective. Suppose {p, l} W (7, q). The equations defining the Wenger graph are equivalent to the statement φ(l) = (1, 0, p 2, p 3, p 4,..., p 7 ) + l 1 (0, 1, p 1, p 2 1, p 3 1,..., p 6 1) φ(p). Therefore, W (7, q) H(7, q). Denote by G q (k) the graph obtained from G q by replacing each vertex v with an independent set S(v) of k vertices, and each edge uv with the complete bipartite graph with parts S(u) and S(v), then writing V (G q (k) = N = kq 4, we have E(G q (k)) = 1 2 k2 (q 5 q 2 ) 1 2 k 3 4 N 5 4 and it can be checked that G q (k) does not contain θ 2k+1,4. This shows ex(n, θ 2k+1,4 ) 1 2 k 3 4 N 5 4 for all k 1. This improves the lower bound ex(n, θ k,4 ) n 5 4 for large enough k, given in [4]. Let F = {C 4, C 6, θ 3,4 ). It is not difficult to show ex(n, F) ( o(1))n 5 4. On the other hand, the graph G q contains neither C 4 nor C 6, and therefore ex(n, F) ( o(1))n 5 4. It is an open question to determine the asymptotic behavior of ex(n, F) or ex(n, θ 3,4 ). References [1] J. A. Bondy, M. Simonovits, Cycles of even length in graphs, J. Combin. Theory Ser. B, 16 (1974), [2] B. Bukh, Z. Jiang, A bound on the number of edges in graphs without an even cycle, arxiv.org/pdf/ v1.pdf. 1 [3] S. M. Cioabă, F. Lazebnik, W. Li, On the Spectrum of Wenger Graphs. J. Combin. Theory Ser. B, 107: (2014) 5 [4] D. Conlon, Graphs with few paths of prescribed length between any two vertices arxiv.org/pdf/ v1.pdf 1, 1, 6 [5] P. Erdős, M. Simonovits, Compactness results in extremal graph theory. Combinatorica 2 (1982), no. 3,

10 10 GRAPHS WITHOUT THETA SUBGRAPHS [6] P. Erdős, A. H. Stone, On the structure of linear graphs. Bulletin of the American Mathematical Society (1946) 52 (12) [7] R. J. Faudree, M. Simonovits, On a class of degenerate extremal graph problems. Combinatorica 3 (1983), no. 1, [8] Z. Füredi, A. Naor, J. Verstraete, On the Turán number for the hexagon, Adv. Math. 203 (2006), no. 2, [9] C. X. He, H. Y. Shan, J. Y. Shao, The existence of even cycles with specific lengths in Wengers graph. Acta Math. Appl. Sin. Engl. Ser, 24: (2008). 5 [10] F. Lazebnik, A. Thomason, Y. Wang, On Some Cycles in Wenger Graphs. Acta Math. Appl. Sin. Engl., to appear. 5 [11] F. Lazebnik, V. Ustimenko, A. J. Woldar, Polarities and 2k-cycle-free graphs. Discrete Math. 197/198 (1999), , 1 [12] F. Lazebnik, V. A. Ustimenko, A J. Woldar, New upper bounds on the order of cages, Electron. J. Combin, 14, R13, (1997), , 2 [13] K. E. Mellinger, D. Mubayi, Constructions of bipartite graphs from finite geometries. J. Graph Theory, 49 (1) (2005), [14] J. A. Verstraete, Extremal problems for cycles in graphs, preprint. 1 [15] R. Viglione, On the diameter of Wenger graphs, Acta Appl. Math. 104 (2008), [16] R. Viglione, Properties of some algebraically defined graphs. Ph.D. Thesis, University of Delaware, [17] R. Wenger, Extremal graphs with no C 4s, C 6s, orc 10s, J. Combin. Theory Ser. B 52 (1991), Department of Mathematics, University of California, San Diego (UCSD), La Jolla, CA , USA address: jverstra@ucsd.edu Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA address: jwilliford@uwyo.edu