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Original citation:
Král', Daniel, Liu, ChunHung, Sereni, JeanSébastien, Whalen, Peter and Yilma,
Zelealem B.. (2013) A new bound for the 2/3 conjecture. Combinatorics, Probability and
Computing, Volume 22 (Number 03). pp. 384393. ISSN 09635483
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A new bound for the 2/3 conjecture
∗
Daniel Kr
´
al’
†
ChunHung Liu
‡
JeanS
´
ebastien Sereni
§
Peter Whalen
¶
Zelealem B. Yilma
k
Abstract
We show that any nvertex 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 suﬃce.
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 aﬃrmative 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 aﬃliation: 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. Email: 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
(FP7/20072013)/ERC grant agreement no. 259385.
‡
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA. Email:
cliu87@math.gatech.edu.
§
CNRS (LORIA), Nancy, France. Email: 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 303320160, USA. Email:
pwhalen3@math.gatech.edu.
k
LIAFA (Universit´e Denis Diderot), Paris, France. Email: 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 rcoloring of the edges of the complete graph, a subset A of the
vertex set cdominates 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 cdominates 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 ﬁxed positive integer t and any 2coloring 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 cdominated by X.
In a more general form, they asked: Given positive integers r, t, and n along with an
rcoloring 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 telement subset
of K
n
? However, in the same paper [7], the authors presented a 3coloring 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 3coloring 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 followup 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 3coloring 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
suﬃcient.
Theorem 2. For any 3coloring 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 cdominates 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
suﬃce to monochromatically dominate a set of size 2n/3 in any 3coloring of the edges of
K
n
.
2
We note that there exist 3colorings 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 3coloring, the probability that an arbitrary pair of vertices monochromatically
dominate more than 5n/9 + o
(
n
)
vertices is o
(
1
)
by Chernoﬀ’s bound.
Our proof of Theorem 2 utilizes the ﬂag 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 diﬃcult 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´antype
problems [25], supersaturation 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 trianglefree 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 3colorings 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 vertexset and
G is the number of vertices of G. Let F
ℓ
be the set of nonisomorphic tricolored graphs
with ℓ vertices, where two tricolored graphs are considered to be isomorphic if they diﬀer
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 ≔ ∪
ℓ∈N
F
ℓ
. Given a tricolored graph σ, we deﬁne F
σ
ℓ
to be the set of tricolored graphs
F on ℓ 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 diﬀers from σ only by a permutation
of the edge colors. The elements of F
σ
ℓ
are usually called σﬂags 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
≔ ∪
ℓ∈N
F
σ
ℓ
.
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 coeﬃcients. 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
H∈F

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 welldeﬁned
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