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Journal ArticleDOI

A New Bound for the 2/3 Conjecture †

01 May 2013-Combinatorics, Probability & Computing (Cambridge University Press)-Vol. 22, Iss: 3, pp 384-393
TL;DR: In this paper, it was shown that any n-vertex complete graph with edges colored with three colors contains a set of at most four vertices such that the number of neighbors of these vertices in one of the colors is at least 2n/3.
Abstract: We show that any n-vertex complete graph with edges colored with three colors contains a set of at most four vertices such that the number of the neighbors of these vertices in one of the colors is at least 2n/3. The previous best value, proved by Erdos, Faudree, Gould, Gyarfas, Rousseau and Schelp in 1989, is 22. It is conjectured that three vertices suffice.

Summary (2 min read)

1 Introduction

  • They further inquired whether 2/3 may be replaced by 1/2.
  • ‡School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA.
  • This author’s work was par- tially supported by the FrenchAgence Nationale de la Rechercheunder referenceanr 10 jcjc 0204 01.
  • The result of Erdőset al. [7] may then be stated as follows.
  • Their proof seems to suggest that Kierstead’s coloring is somehow extremal, giving more credence to the conjecturethat three vertices would suffice to monochromatically dominate a set of size 2n/3 in any 3-coloring of the edges of Kn. 2.

2 Flag Algebras

  • Flag algebras were introduced by Razborov [23] as a tool based on the graph limit theory of Lovász and Szegedy [20] and Borgset al.[5] to approach problems pertaining to extremal graph theory.
  • Standard elementary probability computations [23, Lemma 2.4] show that this multiplication inRF gives rise to a well-defined multiplication in the factor algebraA.
  • Formally, the authors letp(H,H′) be the probability that ν′(V(σ)) together with a randomly chosen subset of|H| − |σ| vertices inV(H′) \ ν′(V(σ)) induce a subgraph that is isomorphic toH through an isomorphismf that preserves the embeddings, that is,ν′ = f ◦ ν.
  • A standard argument (using Tychonoff’s theorem [26]) yields that every sequence of tricolored graphs has a convergent subsequence.
  • Let us now have a closer look at the relation betweenq and qσ.

2.1 Particular Notation Used in our Proof

  • Before presenting the proof of Theorem 2, the authors need to introduce some notation and several lemmas.
  • Recall thatσA, σB andσC, the elements ofF3, are given in Figure 1.

3 Proof of Theorem 2

  • Specifically, the authors first finda umber of flag inequalities 6 by hand and then they combine them with appropriate coeffici nts to obtain a contradiction.
  • The authors define acounterexampleto be a tricolored graph withn vertices such that for every colorc ∈ {1, 2, 3}, each setW of at most four vertices stronglyc-dominates less than 2n/3 vertices ofG.
  • In particular, eitherwj has noc-neighbors insideWj or vj is incident with edges of three distinct colors inG.
  • As the authors shall see, this structural property of counterexamples directly implies their nonexistence, thereby proving Theorem 2. Lemma 6.
  • In particular, the sumw3 + w0, which belongs toRF5, has only non-positive coefficients.

4 Concluding remarks

  • It is natural to ask what bound can be proven for domination with three vertices.
  • Here, it does not seem that the trick the authors used in this paper helps.
  • Now consider the following graphG: start from the disjoint union of a large clique of order 2m with all edges colored 1 and a rainbow triangle.
  • Thus, the average proportion of vertices dominated by triples isomorphic toσC in the graphs (Gk)k∈N is close to 1/2.
  • This phenomenon does not occur for quadruples of vertices.

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Original citation:
Král', Daniel, Liu, Chun-Hung, Sereni, Jean-Sébastien, Whalen, Peter and Yilma,
Zelealem B.. (2013) A new bound for the 2/3 conjecture. Combinatorics, Probability and
Computing, Volume 22 (Number 03). pp. 384-393. ISSN 0963-5483
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A new bound for the 2/3 conjecture
Daniel Kr
´
al’
Chun-Hung Liu
Jean-S
´
ebastien Sereni
§
Peter Whalen
Zelealem B. Yilma
k
Abstract
We show that any n-vertex complete graph with edges colored with three colors
contains a set of at most four vertices such that the number of the neighbors of these
vertices in one of the colors is at least 2n/3. The previous best value, proved by Erd˝os,
Faudree, Gould, Gy´arf´as, Rousseau and Schelp in 1989, is 22. It is conjectured that
three vertices suce.
1 Introduction
Erd˝os and Hajnal [9] made the observation that for a fixed positive integer t, a positive real
ǫ, and a graph G on n > n
0
vertices, there is a set of t vertices that have a neighborhood
of size at least (1 (1 + ǫ)(2/3)
t
)n in either G or its complement. They further inquired
whether 2/3 may be replaced by 1/2. This was answered in the armative by Erd˝os,
This work was done in the framework of LEA STRUCO.
Institute of Mathematics, DIMAP and Department of Computer Science, University of Warwick, Coven-
try CV4 7AL, United Kingdom. Previous aliation: Institute of Computer Science (IUUK), Faculty of
Mathematics and Physics, Charles University, Malostransk´e n´amˇest´ı 25, 118 00 Prague 1, Czech Repub-
lic. E-mail: D.Kral@warwick.ac.uk. The work of this author leading to this invention has received
funding from the European Research Council under the European Union’s Seventh Framework Programme
(FP7/2007-2013)/ERC grant agreement no. 259385.
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA. E-mail:
cliu87@math.gatech.edu.
§
CNRS (LORIA), Nancy, France. E-mail: sereni@kam.mff.cuni.cz. This author’s work was par-
tially supported by the French Agence Nationale de la Recherche under reference anr 10 jcjc 0204 01.
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA. E-mail:
pwhalen3@math.gatech.edu.
k
LIAFA (Universit´e Denis Diderot), Paris, France. E-mail: Zelealem.Yilma@liafa.jussieu.fr.
This author’s work was supported by the French Agence Nationale de la Recherche under reference anr 10
jcjc 0204 01.
1

Faudree, Gy´arf´as and Schelp [7], who not only proved the result but also dispensed with
the (1 + ǫ) factor. They also phrased the question as a problem of vertex domination in a
multicolored graph.
Given a color c in an r-coloring of the edges of the complete graph, a subset A of the
vertex set c-dominates another subset B if, for every y B \ A, there exists a vertex x A
such that the edge xy is colored c. The subset A strongly c-dominates B if, in addition,
for every y B A, there exists a vertex x A such that xy is colored c. (Thus, the two
notions coincide when A B = .) The result of Erd˝os et al. [7] may then be stated as
follows.
Theorem 1. For any xed positive integer t and any 2-coloring of the edges of the complete
graph K
n
on n vertices, there exist a color c and a subset X of size at most t such that all
but at most n/2
t
vertices of K
n
are c-dominated by X.
In a more general form, they asked: Given positive integers r, t, and n along with an
r-coloring of the edges of the complete graph K
n
on n vertices, what is the largest subset B
of the vertices of K
n
necessarily monochromatically dominated by some t-element subset
of K
n
? However, in the same paper [7], the authors presented a 3-coloring of the edges
of K
n
attributed to Kierstead which shows that if r > 3, then it is not possible to
monochromatically dominate all but a small fraction of the vertices with any fixed number
t of vertices. This 3-coloring is defined as follows: the vertices of K
n
are partitioned into
three sets V
1
, V
2
, V
3
of equal sizes and an edge xy with x V
i
and y V
j
is colored i if
1 6 i 6 j 6 3 and j i 6 1 while edges between V
1
and V
3
are colored 3. Observe that, if
t is fixed, then at most 2n/3 vertices may be monochromatically dominated.
In the other direction, it was shown in the follow-up paper of Erd˝os, Faudree, Gould,
Gy´arf´as, Rousseau and Schelp [8], that if t > 22, then, indeed, at least 2n/3 vertices are
monochromatically dominated in any 3-coloring of the edges of K
n
. The authors then ask
if 22 may be replaced by a smaller number (specifically, 3). We prove here that t > 4 is
sucient.
Theorem 2. For any 3-coloring of the edges of K
n
, where n > 2, there exist a color c and
a subset A of at most four vertices of K
n
such that A strongly c-dominates at least 2n/3
vertices of K
n
.
In Kierstead’s coloring, the number of colors appearing on the edges incident with
any given vertex is precisely 2. As we shall see later on, this property plays a central
role in our arguments. In this regard, our proof seems to suggest that Kierstead’s coloring
is somehow extremal, giving more credence to the conjecture that three vertices would
suce to monochromatically dominate a set of size 2n/3 in any 3-coloring of the edges of
K
n
.
2

We note that there exist 3-colorings of the edges of K
n
such that no pair of vertices
monochromatically dominate 2n/3 + O
(
1
)
vertices. This can be seen by realizing that in
a random 3-coloring, the probability that an arbitrary pair of vertices monochromatically
dominate more than 5n/9 + o
(
n
)
vertices is o
(
1
)
by Chernos bound.
Our proof of Theorem 2 utilizes the ag algebra theory introduced by Razborov, which
has recently led to numerous results in extremal graph and hypergraph theory. In the
following section, we present a brief introduction to the flag algebra framework. The
proof of Theorem 2 is presented in Section 3.
We end this introduction by pointing out another interesting question: what happens
when one increases r, the number of colors? Constructions in the vein of that of Kierstead
for example, partitioning K
n
into s parts and using r =
s
2
colors show that the size
of dominated sets decreases with increasing r. While it may be dicult to determine the
minimum value of t dominating a certain proportion of the vertices, it would be interesting
to find out whether such constructions do, in fact, give the correct bounds.
2 Flag Algebras
Flag algebras were introduced by Razborov [23] as a tool based on the graph limit theory of
Lov´asz and Szegedy [20] and Borgs et al. [5] to approach problems pertaining to extremal
graph theory. This tool has been successfully applied to various topics, such as Tur´an-type
problems [25], super-saturation questions [24], jumps in hypergraphs [2], the Caccetta-
H¨aggkvist conjecture [17], the chromatic number of common graphs [14] and the number
of pentagons in triangle-free graphs [12, 15]. This list is far from being exhaustive and
results keep coming [1, 3, 4, 6, 11, 10, 13, 16, 18, 19, 21, 22].
Let us now introduce the terminology related to flag algebras needed in this paper.
Since we deal with 3-colorings of the edges of complete graphs, we restrict our attention
to this particular case. Let us define a tricolored graph to be a complete graph whose
edges are colored with 3 colors. If G is a tricolored graph, then V(G) is its vertex-set and
|G| is the number of vertices of G. Let F
be the set of non-isomorphic tricolored graphs
with vertices, where two tricolored graphs are considered to be isomorphic if they dier
by a permutation of the vertices and a permutation of the edge colors. (Therefore, which
specific color is used for each edge is irrelevant: what matters is whether or not pairs of
edges are assigned the same color.) The elements of F
3
are shown in Figure 1. We set
F
N
F
. Given a tricolored graph σ, we define F
σ
to be the set of tricolored graphs
F on vertices with a fixed embedding of σ, that is, an injective mapping ν from V(σ) to
V(F) such that Im(ν) induces in F a subgraph that diers from σ only by a permutation
of the edge colors. The elements of F
σ
are usually called σ-flags within the flag algebras
3

σ
A
1 2
3
σ
B
1 2
3
σ
C
1 2
3
Figure 1: The elements of F
3
. The edges of color 1, 2 and 3 are represented by solid,
dashed and dotted lines, respectively.
framework. We set F
σ
N
F
σ
.
The central notions are factor algebras of F and F
σ
equipped with addition and multi-
plication. Let us start with the simpler case of F. If H F and H
F
|H|+1
, then p(H, H
)
is the probability that a randomly chosen subset of |H| vertices of H
induces a subgraph
isomorphic to H. For a set F, we define RF to be the set of all formal linear combinations
of elements of F with real coecients. Let A RF and let F be A factorised by the
subspace of RF generated by all combinations of the form
H
X
H
F
|H|+1
p(H, H
)H
.
Next, we define the multiplication on A based on the elements of F as follows. If
H
1
and H
2
are two elements of F and H F
|H
1
|+|H
2
|
, then p(H
1
, H
2
; H) is the probability
that two randomly chosen disjoint subsets of vertices of H with sizes |H
1
| and |H
2
| induce
subgraphs isomorphic to H
1
and H
2
, respectively. We set
H
1
· H
2
X
HF
|
H
1
|
+
|
H
2
|
p(H
1
, H
2
; H)H.
The multiplication is linearly extended to RF. Standard elementary probability compu-
tations [23, Lemma 2.4] show that this multiplication in RF gives rise to a well-defined
multiplication in the factor algebra A.
The definition of A
σ
follows the same lines. Let H and H
be two tricolored graphs in
F
σ
with embeddings ν and ν
of σ. Informally, we consider the copy of σ in H
and we
extend it into an element of F
σ
|H|
by randomly choosing additional vertices in H
. We are
interested in the probability that this random extension is isomorphic to H and the isomor-
phism preserves the embeddings of σ. Formally, we let p(H, H
) be the probability that
ν
(V(σ)) together with a randomly chosen subset of |H| |σ| vertices in V(H
) \ ν
(V(σ))
induce a subgraph that is isomorphic to H through an isomorphism f that preserves the
embeddings, that is, ν
= f ν. The set A
σ
is composed of all formal real linear combina-
tions of elements of RF
σ
factorised by the subspace of RF
σ
generated by all combinations
4

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Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "A new bound for the 2/3 conjecture" ?

The authors show that any n-vertex complete graph with edges colored with three colors contains a set of at most four vertices such that the number of the neighbors of these vertices in one of the colors is at least 2n/3.