talk in Boca Raton in Part of the reason for not publishing the result for such a long time was his hope for an improvement in terms of the size

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1 Embedding arbitrary nite simple graphs into small strongly regular graphs. Robert Jajcay Indiana State University Department of Mathematics and Computer Science Terre Haute, IN Dale Mesner University of Nebraska Department of Mathematics and Statistics Lincoln, NE February 4, 1997 Abstract It is well-known that any nite simple graph? is an induced subgraph of some exponentially larger strongly regular graph? (e.g. [2, 8]). No general polynomial-size construction has been known. For a given - nite simple graph? on v vertices we present a construction of a strongly regular graph? on O(v 4 ) vertices that contains? as its induced subgraph. A discussion is included of the size of the smallest possible strongly regular graph with this property. 1 Introduction and basic concepts. The original idea for the construction introduced in this paper appeared to the second of the authors around He later presented a preliminary short 1

2 talk in Boca Raton in Part of the reason for not publishing the result for such a long time was his hope for an improvement in terms of the size of the strongly regular graph used in the construction. The present paper extends the original result in a manner where we were actually able to show that the construction cannot be improved any further. In the last section we address the possibility of obtaining better bounds by using dierent families of strongly regular graphs. All the graphs we consider are nite simple graphs. Given such a graph? = (V; E) together with a subset S of V, the induced (or vertex-induced) subgraph of? corresponding to S is a subgraph of? which has S as its vertex set and whose edges are those edges of? which have both endpoints in S. A k-regular graph? on v vertices is said to be strongly regular provided it satises the following additional \regularity" properties for some integers and : 1. Each pair of adjacent vertices has exactly common neighbors, and 2. each pair of non-adjacent vertices has exactly common neighbors. In this case, the four-tuple (v; k; ; ) is called the type of the strongly regular graph?. Besides being strikingly nice, strongly regular graphs possess several interesting combinatorial properties and are closely related to combinatorial designs and codes as well as partial geometries (anyone interested in this aspect of strongly regular graphs is referred to [5, 3] which also include all the basic notation). Out of the several well-known families of strongly regular graphs, we will be particularly interested in strongly regular graphs based on Desarguesian ane plane geometries. Assume that n is the order of a nite eld F, and consider the Desarguesian ane plane geometry coordinatized by F. The plane has n 2 points (x; y) which we take as vertices of our strongly regular graph. The geometry contains n 2 lines of the form y = mx + b, m; b 2 F, and n vertical lines; all the lines contain n points. The parameter m is said to be the slope of the line; vertical lines are usually thought of as having an innite slope. All the lines having the same slope m are disjoint and are called a parallel class of slope m. The vertical lines form the parallel class of innite slope. 2

3 Consider a graph? whose vertices are the n 2 points (x; y) of our geometry. Let g be an integer, 2 g n + 1, and choose any g of the n + 1 parallel classes of lines in the geometry. We refer to these as special parallel classes and to the lines in them as special lines or lines with special slopes. Two vertices of the graph? are dened to be adjacent i the line joining them in the geometry is one of the special lines. It is easy to see that any graph? constructed in the above described manner is a strongly regular graph of the type (n 2 ; g(n? 1); n? 2 + (g? 1)(g? 2); g(g? 1)). The above introduced strongly regular graphs based on Desarguesian af- ne planes are members of a more general family of Latin square graphs. 2 The main construction. As mentioned in the abstract, the fact that any nite simple graph can be embedded as an induced subgraph of some strongly regular graph has been known for quite some while. We are aware of several papers that have introduced such a construction, mostly as a byproduct of solving other related problems. Let us mention at least the paper [2] in which the authors present an embedding of arbitrary graphs into Paley graphs. However, in terms of the order v of the given graph?, all the previously known constructions are exponential in v, i.e., the obtained strongly regular graph? is of order O(c v ), for a positive constant c. This is also the case of the recently published article [8] that has been brought to our attention by the referees, and in which the author uses strongly regular graphs based on Steiner systems. Although the general algorithm of this paper is still exponential in v, for the special family of block graphs of triple partial designs the obtained strongly regular graphs are of polynomial order cv 2, for a positive constant c. In the following section, we introduce a construction that is polynomial for all nite simple graphs. Let us rst explain the basic idea of our construction by reproving the existence of a strongly regular graph containing a given graph as an induced subgraph. Theorem 1 If? is a nite simple graph, then there exists a strongly regular graph? which has an induced subgraph isomorphic to?. 3

4 Proof. Suppose? has v vertices and e edges. In an ane plane of suciently large order n, choose a set S of v points such that the v(v? 1)=2 lines joining the points in pairs have distinct slopes. Take the n 2 vertices of the plane as vertices of graph? and choose a one-to-one correspondence associating each vertex of? with a vertex in S. The -images of the endpoints of each edge of? now determine a line in the plane and the e lines so determined are in e distinct parallel classes. Take these as the g = e special parallel classes which dene adjacency in?. Then? will be a strongly regular graph based on ane plane geometry, and the vertex set S will induce a subgraph with an edge corresponding to each edge of?. The (v(v?1)=2)?e pairs of non-adjacent vertices of? determine lines of the plane which have non-special slopes and do not give rise to any edges of?. Thus the subgraph of? induced by the vertices from S does not have any extraneous edges and is isomorphic to?. 2 There is an obvious and quite essential omission in the above proof. Namely, what is the size of \a suciently large" number n that allows one to nd the set S of v points determining v(v? 1)=2 lines all of them having distinct slopes? A simple combinatorial argument can be made to argue the suciency of n of the magnitude of about O(v 3 ) (v being the vertex-size of the original graph?). As a result, the desired graph? would have the order of the polynomial magnitude of about O(v 6 ). In order to further improve this bound, we consider the following concept. Given a nite abelian group G, we say that a subset D = fx 1 ; x 2 ; : : : ; x k g of G is a Sidon set in G provided the set of sums fx i + x j jx i ; x j 2 D; i jg of pairs of elements from D consists of (k(k? 1)=2) + k distinct elements of G (i.e. no two of the sums are equal). The concept of a Sidon set was originally proposed by S. Sidon in 1933 in [6, 7] for the set of positive integers, and we would like to express our thanks to Professor Laszlo Babai who was polite enough to point the reference to us. The idea of Sidon sets can be easily extended to all groups in general (not necessarily abelian), and the interested reader is referred to [1]. Now, let F be a nite eld of n elements again, and let D = fx 1 ; x 2 ; : : : ; x v g be a Sidon set in the nite abelian group (F; +). Take S to be the set f(x 1 ; x 2 1); (x 2 ; x 2 2); : : : ; (x v ; x 2 v)g. We claim that all the lines determined by pairs of points in S have dierent slopes. Indeed, consider two pairs of points from S : (x ; i 1 x2 ); (x ; i1 i2 x2 ) and (x ; i2 i3 x2 ); (x ; i3 i4 x2 ), where at least three of i4 these points are dierent. The slopes of the two lines determined by the pairs 4

5 are : and (x 2 i2? x2 i1 )=(x i2? x i1) = x i 2 + x i 1 (x 2 i4? x2 i3 )=(x i4? x i3) = x i 4 + x i 3 ; and thus they are certainly dierent. The existence of a Sidon set of a suf- ciently large size in the cyclic group (F; +) is guaranteed by the following classical result of Erdos and Turan ([4]): Lemma 1 Let G be a nite cyclic group of order n. Sidon set D of size cn 1=2. Then G contains a This nally implies the following theorem about the size of a strongly regular graph that contains a given nite simple graph. Theorem 2 If? is a nite simple graph on v vertices, then there exists a strongly regular graph? on O(v 4 ) vertices that contains an induced subgraph isomorphic to?. Proof. Let? be a nite simple graph on v vertices, and let q be the smallest prime such that c p q > v (c being the positive constant from Lemma 1). Consider the nite eld F of q elements. While q is large enough to guarantee the existence of a Sidon subset D = fx 1 ; x 2 ; : : : ; x v g of size v inside of (F; +), it is a well known number-theoretical fact that q, the smallest prime larger than v2, belongs to O(v 2 ). The ane plane based c 2 on F has therefore O(v 4 ) points. This is also the order of the resulting strongly regular? obtained by identifying the vertices of? with the elements of f(x 1 ; x 2 1); (x 2 ; x 2 2); : : : ; (x v ; x 2 v)g. 2 3 Notes on the order of a minimal?. As shown in the previous proof, given a graph on v vertices, we can always embed it into a strongly regular graph on O(v 4 ) vertices. Let f(v) denote the smallest positive integer for which every nite simple graph on v vertices can be embedded into some strongly regular graph on at most f(v) vertices. The above construction shows that : f(v) c v 4 ; (1) 5

6 for all v and a positive constant c. The best currently known lower bound on f(v) appears in the recent article [8], where it is shown that : c 0 v 2 f(v); (2) for all v and a positive constant c 0. This leaves one with a relatively small range of possibilities in terms of the magnitude of the function f(v). The authors do not know whether either the upper bound from (1) or the lower bound from (2) can be improved. Although we have not found any general constructions, we nd the existence of a quadratic embedding construction feasible. In what follows, we shall argue, however, that the construction using strongly regular graphs based on Desarguesian ane planes cannot be improved, and thus we have found the best possible construction using strongly regular graphs of this special class. We proceed by contradiction. Let f a (v) denote now the smallest positive integer for which each nite simple graph on v vertices can be embedded into some strongly regular graph based on a Desarguesian ane plane on at most f a (v) vertices. Suppose, to the contrary, that f a (v) is strictly less than O(v 4 ), that is, suppose that f a (v) belongs to o(v 4 ). The number of nite simple graphs of order v is greater than or equal to the number 2 v(v?1)=2 v! On the other hand, the number of non-isomorphic strongly regular p graphs based on a Desarguesian ane plane with f a (v) vertices q fa(v)+1 is 2 (the number of choices of subsets of parallel classes out of the f a (v) + 1 possible slopes). Since the number of induced subgraphs of order v contained in a single graph of order i is at most the combination number C(i; v), the total number of induced subgraphs of order v contained in strongly regular graphs based on Desarguesian ane planes with f a (v) vertices is smaller than or equal to : fa(v) X i=v C(i; v)2 p p i+1 f a (v)c(f a (v); v)2 fa(v)+1 f a(v) v+1 2 v! p fa(v)+1 = (3) 6

7 2 (v+1) ln(fa(v))+ p fa(v)+1 v! If all the nite simple graphs of order v can be embedded into strongly regular graphs based on Desarguesian ane planes with not more than f a (v) vertices then it certainly has to be true that the total number of nite simple graphs of order v in (3) is smaller than or equal to the last expression of inequality (4). This reduces to v(v? 1) 2 (v + 1) ln(f a (v)) + q f a (v) + 1 which contradicts our assumption that f a (v) is o(v 4 ). We can conclude that the construction introduced in this paper cannot be further improved in terms of the size of the strongly regular graph used, and b 1 v 4 f a (v) b 2 v 4 ; for a pair of positive constants b 1 b 2 and all v (i.e. f a (v) belongs to (v 4 )). Notice again that the above obtained result does not exclude the possibility of improving the O(v 4 ) bound from (1) using another family of strongly regular graphs dierent from those based on Desarguesian ane planes. It should be pointed out, however, that any general embedding construction has to involve a family of strongly regular graphs of unlimited parameters and. This is a simple consequence of the fact that any strongly regular? with parameters (n; k; ; ) containing a given nite simple graph? has to satisfy the two inequalities: and maxfn u;v j fu; vg 2 Eg maxfn u;v j fu; vg 62 Eg; where n u;v denotes the number of common neighbors of u and v in?. The last lemma of our paper indicates yet another way to obtain lower bounds on f(v) by means of a simple two-way counting argument. Let? be a nite simple graph (V; E), jv j = v, and let? be any strongly regular graph containing? as an induced subgraph. Let (n; k; ; ) be the type of?, and let us accept the following shorthand notation: d k := X v2v (k? deg(v)) (4) 7

8 d := d := X fu;vg2e X fu;vg62e (? n u;v ) (? n u;v ) Thus, d k ; d and d denote the total \deciency" of? from being k-regular, having common neighbors for each pair of adjacent vertices, and having common neighbors for each pair of non-adjacent vertices, respectively (for instance, d k sums the total number of edges emanating from vertices of? that do not belong to the set E of edges of?). Lemma 2 Let? and? satisfy the above conditions. Then d 2 k 2(d + d ) + d k (n? v) Proof. Consider the set of all triples fv; u; wg of vertices from? such that u; v 2 V, w 62 V, and both fu; wg and fv; wg are edges of?. Denote the number of such \V-shapes" by r. Then, for any pair of vertices u; v of?, the number of \V-shapes" starting and ending at u; v will be equal to either? n u;v or? n uv, depending on whether the u and v are adjacent or nonadjacent. Thus r is clearly equal to d + d. On the other hand, let y w denote the number of vertices of? adjacent to a vertex w of? not belonging to V. Then r is equal to P (y w (y w? 1)=2) = 1 2 (P y 2 w? P y w ), summing through all the n? v vertices of? that do not belong to V. Since P y w represents the number of edges starting in? and ending in the vertices of? that do not belong to?, P y w = d k, and by comparing the two expressions for r we obtain the equality: 1 2 (X y 2 w? d k) = d + d ; and therefore: X y 2 w = 2(d + d ) + d k : (5) Recall now the Cauchy inequality that asserts the following about the numbers y w : (n? v)( X y 2 w)? ( X y w ) 2 0; 8

9 i.e. X y 2 w (X y w ) 2 =(n? v) = d2 k n? v : (6) Combining the identity (5) with the inequality (6), we obtain the desired result. 2 Note on vertex-transitive graphs. Notice that all the strongly regular graphs based on Desarguesian ane plane geometries used in our construction are in fact Cayley graphs and therefore vertex-transitive (i.e. the full automorphism groups of these graphs act transitively on the sets of vertices). Thus, we have unintentionally proved the following result: Every nite simple graph? on v vertices is an induced subgraph of a vertex-transitive (Cayley) graph? on O(v 4 ) vertices. This result, however, is not new. It has been proved in [1], where, by using a less restrictive family of Cayley graphs, the authors have succeeded in constructing Cayley graphs of order O(v 2 ). The graphs obtained in [1] are generally not strongly regular. Note on the automorphism groups of the obtained strongly regular graphs. Upon hearing about the result obtained in our paper, Eric Mendelsohn has raised the following interesting question: Is it possible to embed any nite simple graph? into a strongly regular? such that? preserves the automorphism group of? (i.e. Aut? = Aut?)? Since all the strongly regular graphs based on Desarguesian ane planes have rich automorphism groups, this question would require a completely dierent technique. References [1] L. Babai and V.T. Sos, Sidon sets in groups and induced subgraphs of Cayley graphs, Europ. J. Combinatorics, (1985) 6, [2] A. Blass, G. Exoo and F. Harary, Paley graphs satisfy all rst order adjacency axioms, J. Graph Theory 5, (1981), no. 4,

10 [3] P.J. Cameron and J.H. van Lint, Graphs, codes and designs, London Math. Soc. Lecture Note Series 43, Cambridge University Press. [4] P. Erdos and P. Turan, On a problem of Sidon in additive number theory and some related problems, J. London Math. Soc , [5] J.H. van Lint and R.M. Wilson, A course in combinatorics, Cambridge University Press, (1992), [6] S. Sidon, Ein Satz uber trigonometrische Polynome und seine Anwendungen in der Theorie der Fourier-Reihen, Math. Ann. 106 (1932), 539. [7] S. Sidon, Uber die Fourier Konstanten der Funktionen der Klasse L p fur p > 1, Acta Sci. Math. (Szeged) 7 (1935), [8] V.H. Vu, On the embedding of graphs into graphs with few eigenvalues, J. Graph Theory 22, (1996), no. 2,