DSpace at VNU: An explicit construction of (3, t)-existentially closed graphs

Discrete Applied Mathematics 161 (2013) 1651–1654 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note An explicit construction of (3, t )-existentially closed graphs Le Anh Vinh ∗ University of Education, Vietnam National University, Ha Noi, Viet Nam article info Article history: Received 21 March 2012 Received in revised form 22 September 2012 Accepted 24 December 2012 Available online 26 February 2013 abstract Let n, t be positive integers A t-edge-colored graph G is (n, t )-e.c or (n, t )-existentially closed if for any t disjoint sets of vertices A1 , , At with |A1 | + · · · + |At | = n, there is a vertex x not in A1 ∪· · ·∪ At such that all edges from this vertex to the set Ai are colored by the i-th color In this paper, we give an explicit construction of a (3, t )-e.c graph of polynomial order © 2013 Elsevier B.V All rights reserved Keywords: n-e.c graph Finite field Gauss sum Introduction For a positive integer n, a graph is n-existentially closed or n-e.c if we can extend all n-subsets of vertices in all possible ways More precisely, for every pair of subsets A, B of vertex set V of the graph such that A ∩ B = ∅ and |A| + |B| = n, there is a vertex z not in A ∪ B that joined to each vertex of A and no vertex of B An n-e.c tournament is defined in an analogous way to an n-e.c graph More precisely, a directed graph is n-e.c tournament if for every triple of disjoint subsets A, B and C such that |A| + |B| + |C | = n, there is a vertex z not in A ∪ B ∪ C that has directed edges going to each vertex of A, directed edges coming from each vertex of B, and no arrow to vertices of C From the results of Erdős and Rényi [3], almost all finite graphs are n-e.c Despite this result, until recently, only a few explicit examples of n-e.c graphs have been known for n > See [1] for a comprehensive survey on the constructions of n-e.c graphs and n-e.c tournaments The techniques used in these known constructions are diverse, emanating from probability theory and random graphs, finite geometry, number theory, design theory, and matrix theory This diversity makes the topic of n-e.c graphs both challenging and rewarding More constructions of n-e.c graphs likely remain undiscovered Apart from their theoretical interest, adjacency properties have recently emerged as an important tool in research on real-world networks Several evolutionary random models for the evolution of the web graph and other self-organizing networks have been proposed The n-e.c property and its variants have been used in [2,4] to analyze the graphs generated by the models, and to help find distinguishing properties of the models In [6], the author studied a multicolor version of this property Let n, t be positive integers A t-edge-colored graph G is (n, t )-existentially closed (or (n, t )-e.c.) if for any t disjoint sets of vertices A1 , , At with |A1 | + · · · + |At | = n, there is a vertex x not in A1 ∪ · · · ∪ At such that all edges from this vertex to the set Ai are colored by the i-th color Since the complement of a graph can be viewed as a color class, we can restrict our discussion to the t-edge-coloring of complete graphs Note that the usual definition of n-e.c graphs is the special case of t = For a positive integer N, the probability space Gt (N , 1t ) consists of all t-colorings of the complete graph of order N such that each edge is colored independently by any color with the probability ∗ Tel.: +84 944058588 E-mail address: leanhvinh@gmail.com 0166-218X/$ – see front matter © 2013 Elsevier B.V All rights reserved doi:10.1016/j.dam.2012.12.016 t The author showed [6, Theorem 1.1] that 1652 L.A Vinh / Discrete Applied Mathematics 161 (2013) 1651–1654 almost all graphs in Gt (N , 1t ) have the property (n, t )-e.c as N → ∞ The proof of this theorem is similar to the proof that almost all finite graphs have the n-e.c property (see, for example, [3]) Although this result implies that there are many (n, t )-e.c graphs, it is nontrivial to construct such graphs The author [6, theorem 1.2] constructed explicitly many graphs satisfying this condition Let q be an odd prime power and Fq be the finite field with q elements Let q be a prime power such that t |(q − 1) and ν be a generator of the multiplicative group of the field Fq We identify the color set with the set {0, , t − 1} The graph Pq,t is a graph with vertex set Fq , the edge between two distinct vertices being colored by the ith color if their sum is of the form ν j where j ≡ i mod t For any positive integers n and t, one can show that Pq,t is an (n, t )-e.c graph when q is large enough More precisely, if q is a prime power such that q > 3(t −1)n q1/2 + n2(t −1)n , (1) then Pq,t has the (n, t )-e.c property Note that the main motivation of that work is to construct new classes of n-e.c graphs From any (n, k)-e.c graph, we can obtain an n-e.c graph by dividing the color set into two sets For a positive integer N and < ρ < 1, the probability space G(N , ρ) consists of graphs with vertex set of size N so that two distinct vertices are joined independently with probability ρ It is known that almost all graphs in G(N , ρ) have the n-e.c graphs The above construction ‘‘supports’’ this statement by constructing explicitly n-e.c graphs with edge density ρ for any < ρ < For any positive integers n, t, let f (n, t ) be the order of the smallest (n, t )-e.c graph Since f (n, t ) ≤ q for any q that satisfies the condition (1), we have that f (n, t ) ≤ 9(t −1)n + n2(t −1)n In particular, if n = then f (3, t ) = O(93t ), which is of exponential order The main purpose of this note is to give new explicit constructions of (3, t )-graphs of polynomial order Let p be a prime such that t |(p − 1), Fp is the finite field of p elements, and ν be a generator of the multiplicative group of the field We identify the color set with the set {0, , t − 1} For any d ≥ 2, the graph Gpd ,t is the complete graph with the vertex set Fdp , the edge between two distinct vertices x, y being colored by the ith color if their distance ∥x − y ∥ = (x1 − y1 )2 + · · · + (xd − yd )2 is of the form ν j where j ≡ i mod t (note that our graphs are just Cayley graphs of Fdp ) We prove that Gpd ,t is a (3, t )-e.c graph when p ≥ t and d ≥ As an immediate corollary, f (3, t ) = O(t 30 ), which is of polynomial order However, we not have any speculation on what the smallest order of a (3, t )-e.c graph is Theorem Let d ≥ 5, p be a prime such that p > t and t | (p − 1), then Gpd ,t has the (3, t )-e.c property The (3, t )-e.c property of the graph Gpd ,t We now give a proof of Theorem For any i ∈ {0, , t − 1}, let Vi = {ν j : j ≡ i mod t } ⊂ Fp It suffices to show that for any three distinct points a = (a1 , , ad ), b = (b1 , , bd ), c = (c1 , , cd ) in Fdq and i, j, k ∈ {0, , t − 1}, there is a point x = (x1 , , xd ) ∈ Fdp , x ̸= a, b, c such that ∥x − a∥ ∈ Vi , ∥x − b∥ ∈ Vj and ∥x − c ∥ ∈ Vk Therefore, we only need to show that there exist u ∈ Vi , v ∈ Vj , and w ∈ Vk such that the following system has at least four solutions (in this case, one of these solutions is different from a, b, and c), (x1 − a1 )2 + · · · + (xd − ad )2 = u (2) (x1 − b1 ) + · · · + (xd − bd ) = v (3) (x1 − c1 ) + · · · + (xd − cd ) = w (4) 2 2 For any x = (x1 , , xd ) ∈ Fdp , define ∥x∥ = x21 + · · · + x2d By eliminating x2i ’s from (3) and (4), we get an equivalent system of equations ∥x − a ∥ = u x · (b − a) = (u − v + ∥b∥ − ∥a∥)/2 (5) x · (c − a) = (u − w + ∥c ∥ − ∥a∥)/2, (7) (6) where x · y is the usual dot product between two vectors x and y We first show that the system of two Eqs (6) and (7) has a solution x0 for some choices of u ∈ Vi , v ∈ Vj , and w ∈ Vk We consider two cases Case Suppose that b − a and c − a are linearly independent For any u ∈ Vi , v ∈ Vj , and w ∈ Vk , it is clear that there is a solution x0 to the system of two Eqs (6) and (7) L.A Vinh / Discrete Applied Mathematics 161 (2013) 1651–1654 1653 Case Suppose that b − a and c − a are linearly dependent Since b − a ̸= c − a ̸= 0, c − a = l(b − a) for some l ̸= 0, The two Eqs (6) and (7) have a common solution if we can choose u ∈ Vi , v ∈ Vj , and w ∈ Vk such that u − w + ∥c ∥ − ∥a∥ = l(u − v + ∥b∥ − ∥a∥), or equivalently, w + (l − 1)u − lv = α, (8) where α = ∥c ∥ + (l − 1)∥a∥ − l∥b∥ ∈ Fp Let N = |{(x, y, z ) ∈ F3p : ν k xt + (l − 1)ν i yt − lν j z t = α}| and N ∗ = |{(x, y, z ) ∈ (F∗p )3 : ν k xt + (l − 1)ν i yt − lν j z t = α}| To show that Eq (8) has a solution (u, v, w) ∈ Vi × Vj × Vk , it suffices to show that N ∗ > If x = 0, for any choice of y, we have at most t choices of z such that ν k xt + (l − 1)ν i yt − lν j z t = α This implies that N ∗ ≥ N − 3pt Therefore, we only need to show that N > 3pt For any x ∈ Fp , let ep (x) = e2π ix/p From the orthogonality property of the exponential sum, we have that N = p−1   p x,y,z ∈F s=0 p ep (s(ν k xt + (l − 1)ν i yt − lν j z t − α)), where the inner sum is p if ν k xt + (l − 1)ν i yt − lν j z t = α and zero, otherwise This implies that N = p2 + =p + p−1   p x,y,z ∈F s=1 p p−1 1 p s=1 ep (s(ν k xt + (l − 1)ν i yt − lν j z t − α))  ep (−sα)   ep (sν x ) x∈Fp k t   ep (sν y ) i t y∈Fp   ep (sν z ) j t (9) z ∈Fp Let      t  ep (λx ) , φt = max∗   λ∈Fp x∈F p then it is a basic result of number theory (see, for example [5]) that √ φt ≤ (t − 1) p (10) Putting (9) and (10) together, we have √ N ≥ p2 − (t − 1)3 (p − 1) p > 3pt Hence N ∗ > and we always can choose u ∈ Vi , v ∈ Vj , and w ∈ Vk such that the two Eqs (6) and (7) have a common solution x0 We have shown that in both cases, the system of two Eqs (6) and (7) has a solution x0 for some choices of u ∈ Vi , v ∈ Vj , and w ∈ Vk Let x1 , , xk be a basis of solutions of the system x · (b − a) = x · (c − a) = Note that k = d − if we are in Case 1, and k = d − if we are in Case Then any linear combination x = x0 +λ1 x1 +· · ·+λk xk is a solution of (6) and (7) Substituting a solution of this form into (5), we get a single quadratic equation of d − variables Since d 5, this quadratic equation has at least qd−4 ≥ solutions Theorem follows immediately Remarks and further questions Note that the proof of Theorem only works for d ≥ It is plausible to conjecture that the graphs are (3, t )-e.c for any d ≥ 2, t = We know that Gp2 ,2 is isomorphic to the Paley graph Pp2 It is well known that Pp is n-e.c for any n given that p is sufficiently large, so Gp2 ,2 is (n, 2)-e.c This observation also works for other values of t Another interesting question is to consider other constructions with different partitions of colors We have not, however, known any results for other cases 1654 L.A Vinh / Discrete Applied Mathematics 161 (2013) 1651–1654 References [1] [2] [3] [4] A Bonato, The search for n-e.c graphs, Contrib Discrete Math (2009) 40–53 A Bonato, J Janssen, Infinite limits of copying models of the web graph, Internet Math (2004) 193–213 P Erdős, A Rényi, Asymmetric graphs, Acta Math Acad Sci Hungar 14 (1963) 295–315 J Kleinberg, R Kleinberg, Isomorphism and embedding problems for infinite limits of scale-free graphs, in: Proceedings of ACM–SIAM Symposium on Discrete Algorithms, 2005 [5] W.M Schmidt, Equations Over Finite Fields, in: Lecture Notes in Math., vol 536, Springer-Verlag, Berlin, Heidelberg, New York, 1976 [6] L.A Vinh, On the adjacency properties of colored graphs, Preprint Further reading [1] [2] [3] [4] [5] A Blass, G Exoo, F Harary, Paley graphs satisfy all first-order adjacency axioms, J Graph Theory (1981) 435–439 B Bollobás, A Thomason, Graphs which contain all small graphs, European J Combin (1981) 13–15 R.L Graham, J.H Spencer, A constructive solution to a tournament problem, Canad Math Bull 14 (1971) 45–48 A Kisielewicz, W Peisert, Pseudo-random properties of self-complementary symmetric graphs, J Graph Theory 47 (2004) 310–316 L.A Vinh, A construction of 3-existentially closed graphs using quadrances, Australas J Combin 51 (2011) 3–6  ... is of exponential order The main purpose of this note is to give new explicit constructions of (3, t ) -graphs of polynomial order Let p be a prime such that t |(p − 1), Fp is the finite field of. .. )-e.c graph when p ≥ t and d ≥ As an immediate corollary, f (3, t ) = O(t 30 ), which is of polynomial order However, we not have any speculation on what the smallest order of a (3, t )-e.c graph... follows immediately Remarks and further questions Note that the proof of Theorem only works for d ≥ It is plausible to conjecture that the graphs are (3, t )-e.c for any d ≥ 2, t = We know that Gp2