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

Introduction For a positive integer n, a graph is n-existentially closed or n-e.c.. More precisely, a directed graph is n-e.c.. From the results of Erdős and Rényi [3], almost all finite

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Discrete Applied Mathematics

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An explicit construction of ( 3 , t ) -existentially closed graphs

University of Education, Vietnam National University, Ha Noi, Viet Nam

a r t i c l e i n f o

Article history:

Received 21 March 2012

Received in revised form 22 September

2012

Accepted 24 December 2012

Available online 26 February 2013

Keywords:

n-e.c graph

Finite field

Gauss sum

a b s t r a c t

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, ,A twith|A1| + · · · + |A t| =n, there is a

vertex x not in A1∪· · ·∪A t such that all edges from this vertex to the set A iare 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

1 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>2 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, ,A twith|A1| + · · · + |A t| = n, there

is a vertex x not in A1∪ · · · ∪A t such that all edges from this vertex to the set A i 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=2

For a positive integer N, the probability space G t(N,1

t)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 1t The author showed [6, Theorem 1.1] that

∗Tel.: +84 944058588.

E-mail address:leanhvinh@gmail.com.

0166-218X/$ – see front matter © 2013 Elsevier B.V All rights reserved.

almost all graphs in G t(N,1

t)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 F q be the finite field with q elements Let q be a prime power such that t| (q−1)andνbe a generator of the multiplicative group of the field Fq We identify the color set with the set {0, ,t −1} The graph P q,tis a graph with vertex set Fq, the edge between two distinct vertices being colored by the

ith color if their sum is of the formνj where j ≡ i mod t For any positive integers n and t, one can show that P q,t is an (n,t)-e.c graph when q is large enough More precisely, if q is a prime power such that

then P q,thas 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 0< 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 densityρfor any 0< ρ <1

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 = 3 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, andνbe 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 G p d,tis the complete graph with the vertex set Fd , the edge between two distinct vertices x,y being

colored by the ith color if their distance∥x−y∥ = (x1−y1)2+ · · · + (x d−y d)2is of the formνj where j≡i mod t (note that

our graphs are just Cayley graphs of Fd ) We prove that G p d,tis a(3,t)-e.c graph when p≥t6and d≥5 As an immediate

corollary, f(3,t) = O(t30), which is of polynomial order However, we do not have any speculation on what the smallest order of a(3,t)-e.c graph is

Theorem 1 Let d≥5,p be a prime such that p>t6and t| (p−1), then G p d,t has the(3,t)-e.c property.

2 The(3,t)-e.c property of the graph G p d,t

We now give a proof ofTheorem 1 For any i∈ {0, ,t−1}, let

V i= { νj:j≡i mod t} ⊂Fp.

It suffices to show that for any three distinct points a = (a1, ,a d),b = (b1, ,b d),c = (c1, ,c d) in Fdand

i,j,k ∈ {0, ,t −1}, there is a point x = (x1, ,x d) ∈ Fd,x ̸= a,b,c such that∥x−a∥ ∈ V i,∥x−b∥ ∈ V jand

∥x−c∥ ∈V k Therefore, we only need to show that there exist u∈V i, v ∈V j, andw ∈V ksuch that the following system

has at least four solutions (in this case, one of these solutions is different from a,b, and c),

For any x= (x1, ,x d) ∈Fd, define

∥x∥ =x2+ · · · +x2.

By eliminating x2

i’s from(3)and(4), we get an equivalent system of equations

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 x0for some choices of u∈V i, v ∈V j, andw ∈V k We consider two cases

Case 1 Suppose that b−a and c−a are linearly independent For any u∈V i, v ∈V j, andw ∈V k, it is clear that there is

a solution x to the system of two Eqs.(6)and(7)

Case 2 Suppose that b−a and c−a are linearly dependent Since b−a̸=c−a̸=0,c−a=l(b−a)for some l̸=0,1 The two Eqs.(6)and(7)have a common solution if we can choose u∈V i, v ∈V j, andw ∈V ksuch that

u− w + ∥c∥ − ∥a∥ =l(u− v + ∥b∥ − ∥a∥ ),

or equivalently,

whereα = ∥c∥ + (l−1)∥a∥ −l∥b∥ ∈Fp

Let

N= |{ (x,y,z) ∈F3p: νk x t+ (l−1)νi y t−lνj z t= α}|

and

N∗= |{ (x,y,z) ∈ (F∗p)3: νk x t+ (l−1)νi y t−lνj z t = α}|.

To show that Eq.(8)has a solution(u, v, w) ∈V i×V j×V k , it suffices to show that N∗>0 If x=0, for any choice of y, we have at most t choices of z such thatνk x t+ (l−1)νi y t−lνj z t = α This implies that N∗≥N−3pt Therefore, we only need

to show that N>3pt.

For any x∈Fp , let e p(x) =e2πix/p From the orthogonality property of the exponential sum, we have that

p



x,y,z∈ Fp

p− 1



s= 0

e p(s(νk x t+ (l−1)νi y t−lνj z t− α)),

where the inner sum is p ifνk x t+ (l−1)νi y t−lνj z t= αand zero, otherwise This implies that

p



x,y,z∈Fp

p− 1



s= 1

e p(s(νk x t+ (l−1)νi y t−lνj z t− α))

=p2+1

p

p− 1



s= 1

e p(−sα)





x∈ Fp

e p(sνk

x t)

 



y∈ Fp

e p(sνi

y t)

 



z∈ Fp

e p(sνj

z t)



Let

φt=max

λ∈ F∗









x∈ Fp

e p(λx t)





 , then it is a basic result of number theory (see, for example [5]) that

Putting(9)and(10)together, we have

N≥p2− (t−1)3(p−1) √p>3pt.

Hence N∗ > 0 and we always can choose u∈ V i, v ∈ V j, andw ∈ V ksuch 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 x0for some choices of u∈V i, v ∈V j, andw ∈V k Let x1, ,x kbe a basis of solutions of the system

x· (b−a) =0

x· (c−a) =0.

Note that k=d−2 if we are in Case 1, and k=d−1 if we are in Case 2 Then any linear combination x=x0+ λ1x1+· · ·+ λk x k

is a solution of(6)and(7) Substituting a solution of this form into(5), we get a single quadratic equation of d−2 variables

Since d>5, this quadratic equation has at least q d− 4≥4 solutions.Theorem 1follows immediately

3 Remarks and further questions

Note that the proof ofTheorem 1only works for d≥5 It is plausible to conjecture that the graphs are(3,t)-e.c for any

d≥2,t=2 We know that G p2 , 2is isomorphic to the Paley graph P2 It is well known that P p is n-e.c for any n given that p

is sufficiently large, so G p2 , 2is(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

[1] A Bonato, The search for n-e.c graphs, Contrib Discrete Math 4 (2009) 40–53.

[2] A Bonato, J Janssen, Infinite limits of copying models of the web graph, Internet Math 1 (2004) 193–213.

[3] P Erdős, A Rényi, Asymmetric graphs, Acta Math Acad Sci Hungar 14 (1963) 295–315.

[4] 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] A Blass, G Exoo, F Harary, Paley graphs satisfy all first-order adjacency axioms, J Graph Theory 5 (1981) 435–439.

[2] B Bollobás, A Thomason, Graphs which contain all small graphs, European J Combin 2 (1981) 13–15.

[3] R.L Graham, J.H Spencer, A constructive solution to a tournament problem, Canad Math Bull 14 (1971) 45–48.

[4] A Kisielewicz, W Peisert, Pseudo-random properties of self-complementary symmetric graphs, J Graph Theory 47 (2004) 310–316.

[5] L.A Vinh, A construction of 3-existentially closed graphs using quadrances, Australas J Combin 51 (2011) 3–6.