4 - Moore Graphs. Jacques Verstraëte

Transcription

1 4 - Moore Graphs Jacques erstraëte jacques@ucsd.edu 1 Introduction In these notes, we focus on the problem determining the maximum number of edges in an n-vertex graph that does not contain short cycles. First, some notation. The Zarankiewicz numbers z(m, n, F ) denote the maximum number of edges in an m n bipartite F -free graph. By passing to a maximum cut in a 2n-vertex extremal F -free graph, we observe: Lemma 1. For all n 1, 1 2ex(2n, F ) z(n, n, F ) ex(2n, F ). Let C 2k denote the cycle of length 2k and C g = {C 3, C 4,..., C g }. The girth of a graph G containing a cycle is the length of a shortest cycle in G, so a graph has girth at least g + 1 if it is C g -free. The distance between vertices u and v in a graph G, denoted d G (u, v) is the minimum length of a uv-path in G. Then d G (, ) is a metric on (G). The diameter of a connected graph G is max{d G (u, v) : u, v (G)}. 2 Moore graphs and girth If G is a d-regular graph of girth at least 2g+1, then clearly (G) 1+d g 1 r=0 (d 1) r by counting vertices at distance at most g from v. This lower bound was extended by Alon, Hoory and Linial [1] to graphs of average degree d and girth at least 2g + 1 as follows. We refer to the theorem as the Moore Bound: 1

2 Theorem 2. Let G have n vertices, girth at least g and average degree d 2. Then g d (d 1) i if g is odd n i=0 g (d 1) i if g is even i=0 Equality holds only if G is a d-regular graph of diameter g Proof. Suppose g = 2k + 1. By Theorem 11 (Notes Part 3), for all i {1, 2,..., k}, there are at least nd(d 1) k 1 non-backtracking walks of length i in G. Some vertex v is the start of at least i j=1 d(d 1)j 1 such non-backtracking walks, and all of the walks of length at least 1 are paths with distinct ends in (G)\{v}. Therefore k d(d 1) i n 1, i=0 and this gives the required bound. Now suppose g = 2k. For some edge {u, v} E(G), there exist 2 k 1 i=1 (d 1)i non-backtracking walks of length at most k starting with {u, v}. All of these walks have distinct ends and therefore (d 1) i n 2. k 1 2 i=1 This completes the proof of the lower bounds on n. We leave the case of characterization of equality as an exercise. The graphs which achieve equality in the Moore Bound are called Moore graphs. In the next section we consider the existence problem for Moore graphs. 2.1 Moore graphs of girth five Moore graphs of girth g {3, 4} are clearly complete graphs and complete bipartite graphs. Also, all cycles are Moore graphs. The interesting cases are the existence of d-regular Moore graphs of girth g when d 3 and g 5. For g = 5, the following was proved by Hoffman and Singleton [10], using linear algebra: Theorem 3. If a d-regular Moore graph of girth five exists, then d {2, 3, 7, 57}. 2

3 Proof. If G is a d-regular n-vertex Moore graph of girth five, then every pair of nonadjacent vertices of G are the ends of a unique path of length two in G. Consequently, if A is the adjacency matrix of G and I and J are the n n identity matrix and n n all 1 matrix, then A 2 + A (d 1)I = J. Since the spectrum of J is 0 n 1 1 1, A has only two eigenvalues distinct from d, and they satisfy x 2 + x (d 1) = 0 i.e. x = 1 2 ( 1 ± 4d 3). The total number of such eigenvalues is n 1 = d 2. If those eigenvalues have multiplicities r and s, then r + s = d 2. We also have 0 = tr(a) = tr(λ) = d r 2 ( 1 + 4d 3) + s 2 ( 1 4d 3) where Λ is the diagonal matrix of eigenvalues of A similar to A. If the eigenvalues are irrational, then r = s and so 2r = d 2. The trace gives 0 = d r, and therefore d = 2. If the eigenvalues are rational, then they are integers and 4d 3 = m 2 for some integer m. Now the trace gives 0 = 1 4 (m2 + 3) r 2 (1 + m) (m2 +3) 2 16r 32 (1 m). Simplification gives m 5 + m 4 + 6m 3 2m 2 (32r 9)m 15 = 0, so m 15 and so m {1, 3, 5, 15} which implies d {3, 7, 57}. The unique d-regular Moore graphs of girth five for d = 3 and d = 7 are the Petersen graph and Hoffman-Singleton graph, shown below: Figure 1: Hoffman-Singleton Graph It is not known if there exists a 57-regular Moore graph of girth five; such a graph would have 3250 vertices. Generalizing the ideas in the above proof, Damerell [4] 3

4 showed that for d 3, no d-regular Moore graph of odd girth g 7 exist. In the case of even girth g 6, the Feit-Higman Theorem [18] shows that in general d-regular Moore graphs of girth g do not exist for d 3: Theorem 4. For g, d 3, if a d-regular Moore graph of girth 2g exists, then g {3, 4, 6}. Furthermore, these exist whenever d = q + 1 for some prime power q. Moore graphs of even girth are bipartite graphs, by Theorem 2. The existence of Moore graphs of girth 2g for g {3, 4, 6} is a consequence of the construction of generalized polygons, first established by Tits [17]. From a combinatorial point of view, a generalized k-gon of order q is a (q + 1)-uniform (q + 1)-regular hypergraph with q k 1 + q k q + 1 vertices and no cycles of length at most k 1. The bigraph of such a hypergraph is a Moore graph, since the parameters exactly match those in Theorem 2 and the girth is exactly 2k. It is our intention to give a brief and self-contained combinatorial description of the construction of generalized k-gons for k {3, 4}, which exist only when k {3, 4, 6} by Theorem 4. For this we require the definition of d-dimensional projective space. 2.2 Projective spaces For a vector space, let [ k] denote the set of all k-dimensional subspaces of. Fixing a prime power q 2 and a positive integer n, define [ ] n k q = k 1 i=0 q n i 1 q k i 1. These are called Gaussian binomial coefficients. When q is clear from the context, we write [ [ n k] instead of n ] k. The Gaussian binomial coefficients count k-dimensional q subspaces of an n-dimensional vector space over F q. Lemma 5. If is an n-dimensional vector space over a finite field F q, then [ ] [ ] n =. k k The elements of [ [ 1], [ 2] and 3] are referred to as points, lines and planes respectively. For any n k l and an n-dimensional vector space over F q, let H q [n, k, l] be the hypergraph whose vertex set is ( ) l and whose edge set is the set of ( ) W l such that W is a subspace of of dimension l. The statistics are described by the following lemma, following from Lemma 5: 4

5 Lemma 6. For n k l, the hypergraph H q [n, k, l] is a [ k l ]q -uniform [ ] n l k l -regular q hypergraph with [ ] n l q vertices. The hypergraph H q [n+1, 2, 1] is a combinatorial representation of the n-dimensional projective space of order q, denoted P G(n, q). We refer to this as a projective plane when n = 2. We draw below a picture of the Fano plane P G(3, 2): Figure 2: Fano plane and Heawood graph G 2 [3, 2, 1] We let G q [n, k, l] be the bigraph of H q [n, k, l]. These graphs provide the existence of Moore graphs of girth six: Theorem 7. For any prime power q, G q [3, 2, 1] is a (q + 1)-regular Moore graph of girth six. Proof. The parameters indeed fit Theorem 2 with d = q + 1 and n = q 2 + q + 1, so it only has to be shown that G q [3, 2, 1] has no cycles of length four. If U, W are two-dimensional subspaces of the three-dimensional vector space over F q, then dim(u W ) = 1. It follows that every pair of vertices of [ 2] in Gq [3, 2, 1] have exactly one common neighbor in [ 1], and therefore Gq [3, 2, 1] contains no quadrilaterals. Therefore G q [3, 2, 1] has girth six. Next we come to Moore graphs of girth eight. Here we work in the projective space P G(4, q). The following was proved by Benson [2]: Theorem 8. Let be a 5-dimensional vector space over F q, and let P = {P [ 1] : P P } and let L = {L [ 2] : L U}. Let G be the bipartite graph with parts P and L such that P P is joined to L L if P L. Then G is a (q + 1)-regular Moore graph of girth eight. 5

6 Proof. It is not hard to check that P = L = q 3 +q 2 +q+1 and that G contains no cycles of length four, simply because dim(l L ) 1 for all L, L L. Furthermore, each L L contains [ ] 2 1 = q + 1 elements of P and each P P is contained in q q + 1 elements of L. It remains to show that G has no cycle of length six. Suppose, for a contradiction, that G contains a cycle of length six with vertices P 1, L 1, P 2, L 2, P 3, L 3 in that order, where P i P and L i L. This implies P 1 = L 1 L 2, P 2 L 2 L 3 and P 3 = L 3 L 1. By definition of P, P 1, P 2, P 3 are self-orthogonal, and pairwise orthogonal too. Therefore if W is the subspace of generated by P 1, P 2, P 3, then W W. Now dim(w ) + dim(w ) 5, so we conclude dim(w ) 2. This implies that P i = P j for some i j, a contradiction. Instead of P, which consists of one-dimensional vector spaces generated by vectors x such that x t x = 0, one could take the one-dimensional vector spaces lying on any non-degenerate quadric. We also point out that a similar construction exists for Moore graphs of girth twelve; the details are supplied in [2]. The example for q = 2 is shown below: Figure 3: Benson graph We refer the reader to Beutelspacher and Rosenbaum [3] for an account of projective planes, and van Maldeghem [18] for an account of generalized polygons. It is important to note, however, that P G(2, q) is not the only projective plane of order q, and there are many other planes with different geometric properties than P G(2, q). 6

7 Perhaps the salient open problem in projective geometry is whether there exists a projective plane of order q when q is not a prime power. The Bruck-Ryser Theorem [3] states that if a projective plane of order q exists, and q 1 mod 4 or q 2 mod 4, then q is a sum of two integer squares, which automatically rules out many non-prime powers, specifically, all q such that some prime factor congruent to 3 mod 4 appears an odd number of times in the prime factorization of q. A computer aided proof [11] shows no projective plane of order 10 is possible, but the case q = 12 is still open. The problem of existence of generalized quadrangles and hexagons of order q seems to be even more challenging. 3 Turán Numbers for short even cycles The order of magnitude of ex(n, C 2k ) is generally not known. Erdős and Simonovits [7] conjectured that the order of magnitude n 1+1/k, and this conjecture remains open except when k {2, 3, 5}. The cases k {2, 3, 5} come from the existence of Moore graphs in Theorem 4 as discussed in the last section. We summarise the results as follows: Theorem 9. For k {2, 3, 5} and n = q k + q k for some prime power q, z(n, n, C 2k ) = (q + 1)n. For all n, z(n, n, C 2k ) n 1+1/k. Proof. Let x be the unique positive real number such that n = x k + x k If G is a bipartite graph of girth at least 2k + 2 and average degree d + 1 2, with parts U and of size n, then G contains at least 2nd(d 1) i 1 (oriented) non-backtracking walks of length i for any i 1, by Theorem 11 of the Notes Part 3. In particular, if k is even, then the number on non-backtracking walks of even length 2i k is at least 2n(1+d+d 2 + +d k ). At least 1+d+d 2 + +d k of these walks start at some vertex v (G), and all their ends are distinct other than v, so 1 + d + d d k n. Therefore d x which shows z(n, n, C 2k ) (x + 1)n. The definition of x ensures x n 1/k, and therefore z(n, n, C 2k ) n 1+1/k. On the other hand, if x is a prime power, q, then we have z(n, n, C 2k ) (q + 1)n for k {2, 3, 5}, due to the existence of Moore graphs of girth 2k+2. Therefore z(n, n, C 2k ) = (q+1)n in this case. For general n, we use the fact that if p r is the rth prime number, then p r p r 1. So if q is the largest prime such that n q = q k + q k n, then q x n 1/k. Since z(n, n, C 2k ) z(n q, n q, C 2k ) = (q + 1)n for k {2, 3, 5} 7

8 we find z(n, n, C 2k ) n 1+1/k for k {2, 3, 5}. Therefore z(n, n, C 2k ) n 1+1/k for k {2, 3, 5}. 3.1 Quadrilateral-free graphs From Theorem 9 and ex(2n, C 4 ) z(n, n, C 4 ), one obtains ex(n, C 4 ) ( n 2 )3/2. On the other hand, the Kövari-Sós-Turán Theorem gives ex(n, C 4 ) 1 2 n3/2. The purpose of this section is to show that this upper bound is asymptotically tight, namely ex(n, C 4 ) 1 2 n3/2, and moreover ex(n, C 4 ) can be determined exactly when n = q 2 + q + 1 for some prime power q. The construction is as follows. Let be a three-dimensional vector space over F q. Let ER q be the graph with (ER q ) = [ ] 1 and the edge-set of ER q is the set of pairs {U, W } such that U and W are orthogonal subspaces. In other words, if U = (x, y, z) and W = (u, v, w) then xu+yv+zw = 0. This indeed defines a graph since the equation xu + yv + zw = 0 is symmetric, and these graphs are called Erdős-Rényi polarity graphs. It is straightforward to check that ER q does not have any quadrilaterals, although it contains triangles. Also, ER q has exactly q 2 + q + 1 vertices and every one-dimensional subspace in ( ) 1 that is not orthogonal to itself has degree q + 1 in the graph. A counting argument shows that there are exactly q + 1 self-orthogonal subspaces, which as vertices each have degree q, and so e(er q ) = 1 2 (q + 1)q q2 (q + 1) = 1 2 q(q + 1)2. Once more the distribution of primes shows ex(n, C 4 ) 1 2 n3/2. The following deep theorem of Füredi [8] shows that ex(n, C 4 ) = e(er q ) when n = q 2 + q + 1 and q is a prime power: Theorem 10. Let G be an extremal quadrilateral-free graph with n = q 2 + q + 1 vertices. Then e(g) 1 2 q(q + 1)2, with equality if and only if G = ER q. The main difficulty is in improving the upper bound d(d 1) n 1 for a C 4 -free n- vertex graph of average degree d, since when n = q 2 +q+1 this only gives ex(n, C 4 ) 1 2 (q + 1)n. We remark that the graphs ER q may be described geometrically by polarities of projective planes see [12] for details on polarities. 8

9 3.2 Hexagon-free graphs By Theorem 9, we also see ex(n, C 6 ) ( n 2 )4/3 and ex(n, C 12 ) ( n 2 )6/5. It turns out (see [12]) that polarity graphs exist in these cases too, which supply improved constructions showing ex(n, C 6 ) 1 2 n4/3 and ex(n, C 10 ) 1 2 n6/5. Erdős and Simonovits [7] conjectured that these lower bounds represent the asymptotic value of ex(n, C 6 ) and ex(n, C 10 ), however this conjectures is false, as shown in the following theorem of Füredi, Naor and the author [9]: Theorem 11. 3( 5 2) ( 5 1) 4/3 n4/3 ex(n, C 6 ) 1 12 ( ) n 4/3. The coefficients in the above theorem are roughly and respectively. The situation for cycles of length ten is more complicated see [9] for further details. 3.3 Revisiting girth five We revisit graphs of girth five. By Theorem 3, there exist no d-regular Moore graphs of girth five when d {2, 3, 7, 57}. One may still ask for the determination of ex(n, C 4 ). Theorem 2 and Theorem 9 together give: Corollary 12. ( n 2 )3/2 ex(n, C 4 ) 1 2 n3/2. In particular, Erdős [5] conjectured that the lower bound is asymptotically tight, but this conjecture remains open and no asymptotic improvements are known over the bounds in the above corollary are known. Parsons [16] showed that extremal graphs of girth five are not bipartite by giving a construction of a 2n-vertex graph of girth five with z(n, n, C 4 ) + Θ(n) edges. 3.4 Main open problem Perhaps the main open problem in the area is to show, as conjectured by Erdős and Simonovits [7], that for all k 2, ex(n, C 2k ) = Θ(n 1+1/k ). 9

10 In particular, the case k = 4 is wide open, with the only bounds being ex(n, C 8 ) = O(n 5/4 ) and ex(n, C 8 ) = Ω(n 6/5 ). Generally, the best lower bounds for ex(n, C 2k ) are given by constructions of Lazebnik, Ustimenko and Woldar [13], slightly exceeding density of the constructions of Lubotzky, Phillips and Sarnak [14] and, earlier, Margulis [15]. The current record for all k is ex(n, C 2k ) = Ω(n 1+2/(3k 2) ) in [13]. The next section will be devoted to the proof that ex(n, C 2k ) = O(n 1+1/k ); this is sometimes referred to as the Even Cycle Theorem. References [1] N. Alon, S. Hoory, N. Linial, The Moore bound for irregular graphs, Graphs Combin. 18 (2002), no. 1, [2] C. Benson, Minimal regular graphs of girths eight and twelve, Canad. J. Math [3] A. Beutelspacher, U. Rosenbaum, Projective geometry: from foundations to applications, Cambridge University Press, Cambridge, 1998, 258 pp. [4] R. Damerell, On Moore geometries. II. Math. Proc. Cambridge Philos. Soc. 90 (1981), no. 1, [5] P. Erdős, On sequences of integers no one of which divides the product of two others and on some related problems, Mitt. Forsch. Inst. Math. Mech. Univ. Tomsk 2 (1938) [6] P. Erdős, Some old and new problems in various branches of combinatorics, in: Graphs and combinatorics, (Marseille, 1995); also: Discrete Mathematics (15) (1997), [7] P. Erdős, M. Simonovits, Compactness results in extremal graph theory. Combinatorica 2 (1982), no. 3, [8] Z. Füredi, unpublished manuscript. [9] Z. Füredi, Zoltan, A. Naor, J. erstraete, On the Turán number for the hexagon. Adv. Math. 203 (2006), no. 2, [10] A. Hoffman, R. Singleton, On Moore graphs with diameters 2 and 3. IBM J. Res. Develop

11 [11] C. Lam, L. Thiel, S. Swiercz, The nonexistence of finite projective planes of order 10, Canad. J. Math. 41 (1989), no. 6, [12] F. Lazebnik,. Ustimenko, A. Woldar, Polarities and 2k-cycle-free graphs, Discrete Math. 197/198 (1999), [13] F. Lazebnik,. Ustimenko, A. Woldar, New upper bounds on the order of cages, The Wilf Festschrift (Philadelphia, PA, 1996). Electron. J. Combin. 4 (1997), no. 2, Research Paper 13, approx. 11 pp. [14] A. Lubotzky, R. Phillips, P. Sarnak, P. Ramanujan graphs, Combinatorica 8 (1988), no. 3, [15] G. Margulis, Explicit constructions of graphs without short cycles and low density codes, Combinatorica 2 (1982), no. 1, [16] T. Parsons, Graphs from projective planes, Aeq. Math. (14) (1976), [17] J. Tits, Sur la trialité et certains groupes qui s en déduisent, Inst. Hautes Études Sci. Publ. Math. (2) (1959) [18] H. van Maldeghem, Generalized polygons, Monographs in Mathematics, Birkhäuser fdc/contactforum/vanmaldeghem.pdf. 11