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

Discrete Applied Mathematics 161 (2013) 1651–1654

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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

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 , . . . , 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 = 2.
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:

0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
doi:10.1016/j.dam.2012.12.016

1
.
t

The author showed [6, Theorem 1.1] that


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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 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 density ρ for 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, 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 6 and d ≥ 5. As an immediate
corollary, f (3, t ) = O(t 30 ), 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 > t 6 and t | (p − 1), then Gpd ,t has the (3, t )-e.c. property.

2. The (3, t )-e.c. property of the graph Gpd ,t
We now give a proof of Theorem 1. 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

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 1. 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

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Case 2. 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, 1.
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 ∗ > 0. 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
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 +

2

=p +

p−1
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 ∗ > 0 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) = 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 +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 − 2 variables.
Since d 5, this quadratic equation has at least qd−4 ≥ 4 solutions. Theorem 1 follows immediately.
3. Remarks and further questions
Note that the proof of Theorem 1 only 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 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.


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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. 4 (2009) 40–53.
A. Bonato, J. Janssen, Infinite limits of copying models of the web graph, Internet Math. 1 (2004) 193–213.
P. Erdős, A. Rényi, Asymmetric graphs, Acta Math. Acad. Sci. Hungar. 14 (1963) 295–315.
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[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 5 (1981) 435–439.
B. Bollobás, A. Thomason, Graphs which contain all small graphs, European J. Combin. 2 (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.