Monotone graph limits and quasimonotone graphs

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by  Bela Bollobas

MONOTONE GRAPH LIMITS AND QUASIMONOTONE
GRAPHS
1
1 BE´LA BOLLOBA´S,SVANTEJANSON,ANDOLIVER RIORDAN
0
2
Abstract. The recent theory of graph limits gives a powerful frame-

n workforunderstandingthepropertiesofsuitable(convergent)sequences
a
(G ) of graphs in terms of a limiting object which may be represented
J n
by a symmetric function W on [0,1], i.e., a kernel or graphon. In this
2
contextitisnaturaltowishtorelatespecificpropertiesofthesequence
2
to specific properties of the kernel. Here we show that the kernel is

monotone (i.e., increasing in both variables) if and only if the sequence
]
O satisfies a ‘quasi-monotonicity’ property definedbya certain functional
tending to zero. As a tool we prove an inequality relating the cut and
C
. L1 norms of kernels of the form W1−W2 with W1 and W2 monotone
h that may be of interest in its own right; no such inequality holds for
t general kernels.
a
m
[

1 1. Introduction
v
6 Recently, Lov´asz and Szegedy [20] and Borgs, Chayes, Lov´asz, So´s and
9 Vesztergombi (see, e.g., [5]) developed a rich theory of graph limits, as-
2
sociating limit objects to suitable sequences (G ) of (dense) graphs with
4 ν
1. |Gν| → ∞, where |Gν| denotes the number of vertices of Gν. The basics of
this theory are outlined in Section 2below; see also Diaconis and Janson [8].
0
1 These graph limits (which are not themselves graphs) can be represented
1 in several different ways; perhaps the most important is that every graph
:
v limit can be represented by a kernel (or graphon) on [0,1], i.e., a symmetric
i measurable function W : [0,1]2 [0,1]. However, this representation is
X
→
in general not unique, see e.g. [20, 4, 8, 3]. More generally, kernels can be
r
a defined on any probability space, see Section 2.
We use Γ to denote an arbitrary graph limit, and write Γ for the graph
W
limit defined by a kernel W. We say that two kernels W and W′ are equiva-
lent ifthey definethesamegraphlimit, i.e., if ΓW = ΓW′. WewriteGν Γ
→
when the sequence (G ) converges to Γ (see [20], [5] and Section 2 below for
ν
definitions); if Γ is represented by a kernel W, i.e., if Γ =Γ , we also write
W
G W.
ν
→
Date: 21 January, 2011.
2000 Mathematics Subject Classification. 05C99.
The first author’s research was supported in part by NSF grants CNS-0721983, CCF-
0728928 and DMS-0906634, and AROgrant W911NF-06-1-0076.
Part of this research was carried when SJ visited the Isaac Newton Institute, Cam-
bridge, duringtheprogramme Stochastic Processes in Communication Sciences, 2010.
1
2 BE´LABOLLOBA´S,SVANTEJANSON,ANDOLIVERRIORDAN
Following [8], we denote the set of all graph limits by , and note that
∞
U
is a compact metric space. Another version of the important compact-
∞
U
ness property for graph limits is that every sequence (G ) of graphs with
ν
G has a convergent subsequence, i.e., a subsequence converging to
ν
| | → ∞
some Γ .
∞
∈ U
Given a suitable class of graphs, it seems interesting to study thegraph
F
limits of , i.e., the set of graph limits arising as limits of sequences of
F
graphs in . One interesting example is the class of threshold graphs, which
F
has several different characterizations, see e.g. [23]. One of them is the
monotonicity property of the neighbourhoods N(v) of the vertices:
There exists a (linear) ordering of the vertices such that
≺
if v w, then N(v) v,w N(w) v,w . (1.1)
≺ \{ } ⊆ \{ }
The graph limits of threshold graphs were studied by Diaconis, Holmes
and Janson [7] (see also [21]), who showed that they are exactly the graph
limits that can be represented by kernels W that take values in 0,1 only
{ }
and are increasing, in that
W(x ,y ) W(x ,y ) if 0 x x 1, 0 y y 1. (1.2)
1 1 2 2 1 2 1 2
≤ ≤ ≤ ≤ ≤ ≤ ≤
Inother words, W is the indicator function of a symmetricincreasing subset
of [0,1]2. (In this paper, ‘increasing’ should always be interpreted in the
weak sense, i.e., as ‘non-decreasing’.) Moreover, the representation by such
a W is unique, if, as is usual, we identify functions that are equal a.e.
Note that the monotonicity properties in (1.1) and (1.2) are obviously
related; this is perhaps best seen if (1.1) is rewritten as a monotonicity
property of the adjacency matrix of the graph (with some exceptions at the
diagonal), so even without the detailed technical study in [7], the condition
(1.2) should not be surprising.
Increasing and decreasing kernels define the same set of graph limits, by
the change of variables x 1 x. Hence we shall talk about monotone
7→ −
kernels rather than increasing kernels, but for simplicity (and without loss
of generality) we consider only increasing ones, so in this paper ‘monotone’
is regarded as synonymous with ‘increasing’.
Themainpurposeofthepresentpaperistostudythelargerclassofgraph
limits represented by arbitrary monotone kernels (taking any values in [0,1],
rather than just the values 0 and 1), and the corresponding sequences of
graphs. We shall also study analytic properties of monotone kernels them-
selves.
Definition. Let be the set of monotone kernels on [0,1], i.e., the set of
↑
W
all symmetric measurable functions W :[0,1]2 [0,1] that satisfy (1.2).
→
Let be the corresponding class of graph limits, i.e., the class of graph
↑
U
limits thatcan berepresentedas Γ for someW . We call thesegraph
W ↑
∈ W
limits monotone.
By definition, every monotone graph limit can be represented by a mono-
tone kernel W on [0,1], butnote that a monotone graph limit may also have
MONOTONE GRAPH LIMITS AND QUASIMONOTONE GRAPHS 3
many representations by non-monotone kernels. For example, a monotone
kernel can be rearranged by an arbitrary measure-preserving bijection from
[0,1] to itself, which will in general destroy monotonicity.
The classes of monotone kernels and of monotone graph limits
↑ ↑
W U
are studied in Section 4. We show there that is a compact subset of
↑
W
L1([0,1]2), and that is a compact subset of . In addition, we consider
↑ ∞
U U
monotone kernels defined on other (ordered) probability spaces, showing
that each such kernel is equivalent to a monotone kernel on [0,1], so the
class is not enlarged by allowing arbitrary probability spaces.
↑
U
Definition. A sequence (G ) of graphs with G is quasimonotone
ν ν
| | → ∞
if it converges to the set , in the sense that each convergent subsequence
↑
U
has as its limit a graph limit in . In this case we will also say that (G )
↑ ν
U
is a sequence of quasimonotone graphs.
In particular, a sequence (G ) converging to a graph limit in is quasi-
ν ↑
U
monotone. Note that it makes no formal sense to ask whether an individual
graph is quasimonotone; just as for quasirandomness, quasimonotonicity is
a property of sequences of graphs.
Example 1.1 (Thresholdgraphsarequasimonotone). Asnotedabove, each
convergent sequence of threshold graphs converges to a limit represented by
a 0/1-valued kernel W . Hence every sequence of threshold graphs
↑
∈ W
(with orders tending to ) is quasimonotone.
∞
Example1.2(Quasirandomgraphsarequasimonotone). Quasirandomgraphs
wereintroducedbyThomason[25,26]assequences(G )ofgraphsthathave
ν
certain properties typical of random graphs. A number of different such
properties turn out to be equivalent, and there are thus many equivalent
characterizations, see Chung, Graham and Wilson [6]. Another characteri-
zation, found by Lov´asz and Szegedy [20], is that a sequence (G ) is quasir-
ν
andom if and only if it converges to a graph limit represented by a constant
kernel W(x,y) = p, for some p [0,1]. (See also [19] and [13].) Since a con-
∈
stant function is monotone, W , and thus every quasirandom sequence
↑
∈ W
of graphs is quasimonotone.
Example 1.3 (Random graphs are quasimonotone). The sequence of ran-
dom graphs G(ν,p) with some fixed p [0,1] and ν = 1,2,... (coupled
∈
in the natural way for different ν) is a.s. quasirandom, and thus a.s. quasi-
monotone.
Ourmainresult(Theorem1.5below)isthatquasimonotonegraphscanbe
characterizedbyaweakeningof (1.1). Asistypicalforconditionsconcerning
convergence to graph limits, this weakening involves taking averages over
subsetsof thevertex set V, ratherthan imposingacondition forall vertices,
and allows for a small ‘error’, making the condition asymptotic.
Given a graph G with vertex set V = V(G), a vertex v of G and a subset
A of V, let
e(v,A) := N(v) A = w A: w v
| ∩ | |{ ∈ ∼ }|
4 BE´LABOLLOBA´S,SVANTEJANSON,ANDOLIVERRIORDAN
denote the number of edges from v to A.
Let x denote the positive part of x, i.e., max x,0 . Writing n := G =
+
{ } | |
V , given a (linear) order on V and a subset A V, define
| | ≺ ⊆
1
Ω (G, ,A) := e(v,A w ) e(w,A v ) (1.3)
0 ≺ n3 \{ } − \{ } +
v≺w
X(cid:0) (cid:1)
1
= e(v,A v,w ) e(w,A v,w ) , (1.4)
n3 \{ } − \{ } +
v≺w
X(cid:0) (cid:1)
Ω (G, ) := maxΩ (G, ,A), and (1.5)
0 0
≺ A⊆V ≺
Ω (G) := minΩ (G, ). (1.6)
0 0
≺ ≺
In the last line the minimum is taken over all n! orders on V. The normal-
ization by n3 ensures that 0 Ω < 1. In fact, Ω < 1/2, and this bound
0 0
≤
can be improved further, but this is not important for our purposes since
we are interested in small values of Ω .
0
Note that Ω (G) = 0 if and only if there exists an order such that
0
≺
Ω (G, ,A) = 0 for every A, i.e., e(v,A v,w ) e(w,A v,w ) for all A
0
≺ \{ } ≤ \{ }
andv w, whicheasily isseentobeequivalent to(1.1), givingthefollowing
≺
result.
Proposition 1.4. A graph G is a threshold graph if and only if Ω (G) =
0
0. (cid:3)
Note that Ω is not intended as a measure of how far a graph is from
0
being a threshold graph (for such a measure, see Section 8). Rather, we
may think (informally!) of a typical quasimonotone graph as being similar
to a random graph in which edges are independent, and the probability p
ij
of an edge ij is increasing in i and in j. In such a graph, one cannot expect
the neighbourhoods of different vertices to be even approximately nested.
But one can expect that for all ‘large’ sets A of vertices, for most i < j,
e(i,A) will be smaller than (or at least not much larger than) e(j,A). The
ideais thatasmallvalueof Ω (G)detects thisphenomenon,withoutrelying
0
on any given labelling of the vertices.
Some variations of the functional Ω will be defined in Section 3, where
0
we shall show that they are asymptotically equivalent for our purposes.
Our main result is the following, proved in Section 7. (All unspecified
limits in this paper are taken as ν .)
→ ∞
Theorem 1.5. Let (G ) be a sequence of graphs with G . Then
ν ν
| | → ∞
(G ) is quasimonotone if and only if Ω (G ) 0.
ν 0 ν
→
We state a special case separately.
Theorem 1.6. Let (G ) be a sequence of graphs with G , and sup-
ν ν
| | → ∞
pose that (G ) is convergent, i.e., G Γ for some graph limit Γ .
ν ν ∞
→ ∈ U
Then Γ if and only if Ω (G ) 0.
↑ 0 ν
∈ U →
MONOTONE GRAPH LIMITS AND QUASIMONOTONE GRAPHS 5
We give several results on monotone graph limits in Sections 4–6. These
includeacharacterizationintermsofafunctionalΩ(W)forkernels,analogu-
ous to Ω for graphs. Along the way we prove some results about monotone
0
kernels that may beof interest in their own right. For example, on functions
that may be written as the difference between two monotone kernels, the
L1 norm and the cut norm may be bounded in terms of each other; see
Theorem 5.5.
Remark 1.7. Lov´asz and Szegedy [22] have studied the class of graph lim-
its represented by 0/1-valued kernels (and the corresponding graph proper-
ties); with a slight variation of their terminology we call such graph limits
random-free. In contrast to the monotone case, it can be shown that every
representing kernel of a random-free limit is a.e. 0/1-valued; see [14]. It
follows that the graph limits that are both monotone and random-free are
exactly the threshold graph limits.
InSection 8,weconsider thefunctionalobtained bytakingthesupremum
over A inside the sum in (1.3) instead of outside as in (1.5). We shall show
that this stronger functional characterizes convergence to threshold graph
limitsinstead ofmonotonegraphlimits; wecall thecorrespondingsequences
of graphs quasithreshold.
1.1. A problem. The convergence G Γ of a sequence (G ) of graphs to
ν ν
→
a graph limit Γ can be expressed using the homomorphism numbers t(F, ):
·
G Γ if and only if t(F,G ) t(F,Γ) for every fixed graph F; see e.g.
ν ν
→ →
[20], [5], [8] for definitions and further results. In particular, the graph limit
Γ is characterized by the family (t(F,Γ)) . The families (t(F,Γ)) that
F F
appear are characterized algebraically by Lov´asz and Szegedy [20].
Problem 1.8. Characterize the families (t(F,Γ)) that appear for Γ .
F ↑
∈ U
Therestofthispaperisorganizedasfollows. Inthenextsectionwereview
some basic properties of the cut metric that we shall rely on throughout the
paper. In Section 3 we introduce some variants of the functional Ω for
0
graphs. In Section 4 we define analogous functionals for kernels and state
several key properties; these are proved in the next two sections, and then
our main results are deduced in Section 7. Finally, in Section 8 we discuss
related functionals characterizing quasithreshold graphs.
2. Kernels and graph limits
Westateheresomestandarddefinitionsandresultsthatweshalluselater
in the paper. For proofs and further details, see e.g. Borgs, Chayes, Lov´asz,
So´s and Vesztergombi [5], Bollob´as and Riordan [3], or Janson [12, 14].
Let ( , ,µ) be a probability space; for simplicity, we will usually abbre-
S F
viate the notation to or ( ,µ).
S S
A kernel (or graphon) on is a symmetric measurable function 2
S S →
[0,1]. We let ( ) denote the set of all kernels on .
W S S
6 BE´LABOLLOBA´S,SVANTEJANSON,ANDOLIVERRIORDAN
If W is an integrable function on 2, we define its cut norm by
S
W (cid:3) := sup W(x,y)f(x)g(y)dµ(x)dµ(y) , (2.1)
k k kfk∞,kgk∞≤1(cid:12)ZS2 (cid:12)
where ∞ denotes the norm(cid:12)(cid:12) in L∞. In other words, the su(cid:12)(cid:12)premum in
k · k
(2.1) is taken over all (real-valued) functions f and g with values in [ 1,1].
−
(Several other versions exist, which are equivalent within constants.) By
considering the supremum over f with g fixed, and vice versa, it is easy to
see that the supremum is unchanged if we restrict f and g to take values in
1 , so we have
{± }
W (cid:3) = sup W(x,y)f(x)g(y)dµ(x)dµ(y) . (2.2)
k k f,g:S→{±1}(cid:12)ZS2 (cid:12)
(cid:12) (cid:12)
This norm defines a metric W(cid:12)1 W2 (cid:3) for kernels on the same(cid:12)probability
k − k
space ; as usual, we identify kernels that are equal a.e.
S
The cut norm may be used to define another (semi)metric δ(cid:3), the cut
metric, as follows. If ϕ : is a measure-preserving map between two
1 2
S → S
probability spaces and W is a kernel on , we let Wϕ be the kernel on
2 1
S S
defined by Wϕ(x,y) := W ϕ(x),ϕ(y) . Let W be a kernel on a probability
1
space and W a kernel on a possibly different probability space . Then
1 2 2
S (cid:0) (cid:1) S
δ(cid:3)(W1,W2):= ϕi1n,ϕf2kW1ϕ1 −W2ϕ2k(cid:3), (2.3)
where the infimum is taken over all couplings (ϕ ,ϕ ) of and , i.e.,
1 2 1 2
S S
over all pairs of measure-preserving maps ϕ : and ϕ :
1 3 1 2 3 2
S → S S → S
fromathirdprobability space 3. Itis notdifficulttoverify that δ(cid:3) satisfies
S
the triangle inequality (see e.g. [14]), but note that δ(cid:3)(W1,W2) may be 0
ϕ
even if W = W , for example if W = W for some measure-preserving
1 6 2 1 2
ϕ: 1 2. Hence, δ(cid:3) is really a semimetric (but is usually called a metric
S → S
for simplicity).
Note thatδ(cid:3)(W1,W2)isdefinedforkernels ondifferentspaces. Moreover,
itisinvariantundermeasure-preservingmaps: δ(cid:3)(W1ϕ1,W2ϕ2) =δ(cid:3)(W1,W2)
for any measure-preserving maps ϕ : ′ , k = 1,2.
k Sk → Sk
Although we allow couplings (ϕ ,ϕ ) defined on an arbitrary third space
1 2
, in (2.3) it suffices to consider the case when = , with a
3 3 1 2
S S S × S
measureµ having marginals µ and µ , taking for ϕ and ϕ the projections
1 2 1 2
π : , k = 1,2. In fact, for an arbitrary coupling (ϕ ,ϕ )
k 1 2 k 1 2
S × S → S
defined on a space ( ,µ ), the mapping (ϕ ,ϕ ): maps µ to
3 3 1 2 3 1 2 3
S S → S ×S
a measure µ on with the right marginals, and it is easily seen that
1 2
Wϕ1 Wϕ2 (cid:3) S= ×WSπ1 Wπ2 (cid:3).
k 1 − 2 k k 1 − 2 k
Although this will be of much lesser importance, we also define the cor-
responding rearrangement-invariant version of the L1 distance:
δ (W ,W ) := inf Wϕ1 Wϕ2 . (2.4)
1 1 2 ϕ1,ϕ2k 1 − 2 kL1(S32)
The coupling definition (2.3) of the cut metric is valid for all and ,
1 2
S S
butincommonspecialcasesitispossible,andoftenconvenient, touseother,
MONOTONE GRAPH LIMITS AND QUASIMONOTONE GRAPHS 7
equivalent, definitions. For example, if = = [0,1] (equipped with the
1 2
S S
Lebesguemeasure, as always), then as shown by Borgs, Chayes, Lov´asz, So´s
and Vesztergombi [5, Lemma 3.5],
ϕ
δ(cid:3)(W1,W2) := inϕfkW1−W2 k(cid:3), (2.5)
taking the infimum over all measure-preserving bijections [0,1] [0,1].
→
We say that two kernels W1 and W2 are equivalent if δ(cid:3)(W1,W2) = 0.
The set of equivalence classes is thus a metric space with the metric δ(cid:3).
A central result [20, 5] is that these equivalence classes are in one-to-one
correspondencewiththegraphlimits. Inotherwords, each kernelW defines
a graph limit Γ , every graph limit can be represented by a kernel in this
W
way, and two kernels define the same graph limit if and only if they are
equivalent. Thus, the cut metric defines the same notion of equivalence
as the one mentioned in the introduction. Furthermore, W and W are
1 2
equivalent if and only if δ (W ,W )= 0, see e.g. [14].
1 1 2
Every kernel is equivalent to a kernel on [0,1], so it suffices to consider
suchkernels. (Weshallnotusethisrestictioninthepresentpaper,however.)
One manifestation of the connection between graph limits and kernels is
the following: If G is a graph with vertices labelled 1,2,...,n, let A (i,j) :=
G
1 i j define its adjacency matrix, and let
{ ∼ }
W (x,y) := A nx , ny .
G G
⌈ ⌉ ⌈ ⌉
This defines a kernel W on [0,1] (or r(cid:0)ather on (0(cid:1),1], which is equivalent).
G
A sequence of graphs with G converges to the graph limit Γ = Γ
ν W
| | → ∞
if and only if δ(cid:3)(WGν,W)→ 0.
Note that W depends on the labelling of the vertices of G, but only in
G
a rather trivial way, and different labellings yield equivalent kernels. Here,
in the study of monotone kernels, the ordering is relevant. If G is a graph
with a given order on V, we therefore define W = W as above, but
G G,≺
≺
using the labelling of the vertices with 1 2 , ignoring the original
≺ ≺ ···
labelling, if any.
3. Further measures of quasimonotonicity
In Section 1 we defined a functional Ω that measures, in an averaged
0
sense, how far the adjacency matrix of a graph is from being monotone.
There are several natural variations of the definition; we shall concentrate
on two.
Firstly, in (1.3) and (1.4), we were careful to exclude v and w from the
set A; this had the advantage of making Ω (G) exactly zero when G is a
0
threshold graph. But most of the time it is more convenient not to do this.
Instead, we consider
1
Ω (G, ,A) := e(v,A) e(w,A) , (3.1)
1 ≺ n3 − +
v≺w
X(cid:0) (cid:1)
8 BE´LABOLLOBA´S,SVANTEJANSON,ANDOLIVERRIORDAN
which differs from (1.4) in that we count all edges into A, and not just the
edges into A v,w . This changes each edge count by at most 1, so
\{ }
Ω (G, ,A) Ω (G, ,A) < 1/n. (3.2)
0 1
| ≺ − ≺ |
As in (1.5) and (1.6), we set
Ω (G, ) := maxΩ (G, ,A), and (3.3)
1 1
≺ A⊆V ≺
Ω (G) := minΩ (G, ). (3.4)
1 1
≺ ≺
Before turning to our second variant, let us note a basic property of Ω .
0
Let e(v,A) denote the number of edges from v to A in the complement Gc
of G. If v / A, then e(v,A) = A e(v,A). Hence, for any v, w and A,
∈ | |−
e(w,A v,w ) e(v,A v,w ) = e(v,A v,w ) e(w,A v,w ).
\{ } − \{ } \{ } − \{ }
From (1.4) it follows that Ω (Gc, ,A) = Ω (G, ,A), where, naturally,
0 0
≻ ≺
denotes the reverse of the order . Thus Ω (Gc, ) = Ω (G, ) and
0 0
≻ ≺ ≻ ≺
Ω (Gc) = Ω (G).
0 0
For Ω one can show similarly, or deduce using (3.2), that Ω (Gc)
1 1
| −
Ω (G) 2/n, say.
1
| ≤
Despite the above symmetry property of Ω , the following ‘locally sym-
0
metrized’ version of the definition turns out to have technical advantages.
Given a graph G, an order on V(G), and A V(G), set
≺ ⊆
Ω (G, ,A) := Ω (G, ,A)+Ω (G, ,V A), (3.5)
2 1 1
≺ ≺ ≺ \
Ω (G, ) := maxΩ (G, ,A) (3.6)
2 2
≺ A⊆V ≺
and
Ω (G) := minΩ (G, ). (3.7)
2 2
≺ ≺
Of course, we could define a corresponding symmetrization of Ω , but we
0
shall not bother.
It is easily seen that all our functionals Ω take values in [0,1] (in fact, in
j
[0, 1)). We have the following relations.
2
Lemma 3.1. If G is a graph with G = n, then
| |
Ω (G) Ω (G) < 1/n, (3.8)
0 1
| − |
and
Ω (G) Ω (G) 2Ω (G). (3.9)
1 2 1
≤ ≤
Consequently, if(G )isasequenceofgraphs with G , thenΩ (G )
ν ν j ν
| |→ ∞ →
0 for some j if and only if this holds for all j = 0,1,2.
Proof. The inequality (3.8) is immediate from (3.2).
The definition (3.5) implies that
Ω (G, ) Ω (G, ) 2Ω (G, ), (3.10)
1 2 1
≺ ≤ ≺ ≤ ≺
which in turn implies (3.9). (cid:3)
MONOTONE GRAPH LIMITS AND QUASIMONOTONE GRAPHS 9
Remark 3.2. Instead of summing in (1.4) or (3.1), in analogy with the
standard definition of ε-regular partitions (see e.g. [2, Section IV.5]), we
may count the number of ‘bad’ pairs (v,w) of vertices v w where the
≺
differencee(v,A) e(w,A) is larger than εn, forsome smallε. Thissuggests
−
the following definition: with an order on the vertex set V, n := V , and
≺ | |
A a subset of V, set
Ω′(G, ,A) := inf ε > 0 : v w :e(v,A) > e(w,A)+εn εn2 ,
1 ≺ ≺ ≤
n (cid:12)(cid:8) (cid:9)(cid:12) o
and define Ω′(G) by taking t(cid:12)he maximum over A with fixe(cid:12)d, and then
1 ≺
minimizingover . Itisastandardobservationthatifx ,...,x takevalues
1 a
≺
in [0,b], then x εab implies that there are at least εa/2 of the x that
i i ≥ i
areatleastεb/2, andthatifatleastεaofthex areatleastεb, thenthesum
i
is at least ε2aPb. Using this it is easy to check that Ω and Ω′ are bounded
1 1
by suitable functions of each other. In fact, it turns out that
1Ω (G) Ω′(G) Ω (G)1/2.
2 1 ≤ 1 ≤ 1
We can also define corresponding modifications of the other Ω .
j
Remark 3.3. Proposition 1.4 says that a graph G is a threshold graph if
and only if Ω (G) = 0. This does not hold for Ω ; in fact, if G contains an
0 1
edge vw, with v w, then Ω (G, , w ) n−3e(v, w ) = n−3 by (3.1);
1
≺ ≺ { } ≥ { }
hence Ω (G) n−3 unless G is empty. Consequently, Ω (G) > 0 for every
1 1
≥
non-empty graph G. On the other hand, Proposition 1.4 and Lemma 3.1
show that Ω (G) 1/n for every threshold graph.
1
≤
We defined each Ω (G) by taking the minimum of Ω (G, ) over all pos-
j j
≺
sible orderings of the vertices. As the next lemma shows, for Ω , ordering
2
≺
the vertices by their degrees d(v) := e(v,V) (resolving ties arbitrarily) is
optimal. This is the main reason for considering Ω .
2
Lemma 3.4. Let < be an order on V such that v < w = d(v) d(w).
⇒ ≤
Then Ω (G) = Ω (G,<).
2 2
Proof. The inequality Ω (G) Ω (G,<) is immediate from the definition
2 2
≤
(3.7), so it suffices to prove the reverse inequality.
Let beany order on V. If v <w, then e(v,V) = d(v) d(w) = e(w,V)
≺ ≤
and thus, for A V,
⊆
e(v,A) e(w,A) = e(v,V) e(w,V)+e(w,V A) e(v,V A)
− − \ − \
e(w,V A) e(v,V A). (3.11)
≤ \ − \
Letf(v,w,A) := e(v,A) e(w,A) andg(v,w,A) := f(v,w,A)+f(v,w,V
− + \
A). By (3.11), if v <w, then f(v,w,A) f(w,v,V A) and thus
(cid:0) (cid:1) ≤ \
g(v,w,A) f(w,v,V A)+f(w,v,A) = g(w,v,A). (3.12)
≤ \
10 BE´LABOLLOBA´S,SVANTEJANSON,ANDOLIVERRIORDAN
Using (3.12) for v < w with v w, we obtain
≻
1
Ω (G,<,A) := g(v,w,A)
2 n3
v<w
X
1 1
= g(v,w,A)+ g(v,w,A)
n3 n3
v<w v<w
vX≺w vX≻w
1 1
g(v,w,A)+ g(w,v,A)
≤ n3 n3
v<w w>v
vX≺w wX≺v
1
= g(v,w,A) = Ω (G, ,A).
n3 2 ≺
v≺w
X
Hence, by (3.6), Ω (G,<) Ω (G, ). Since is arbitrary, this yields
2 2
Ω (G,<) = Ω (G). ≤ ≺ ≺ (cid:3)
2 2
As an immediate consequence of Lemmas 3.4 and 3.1, we have the follow-
ing result for Ω .
1
Corollary 3.5. Let < be an order on V such that v < w = d(v) d(w).
⇒ ≤
Then Ω (G) Ω (G,<) 2Ω (G).
1 1 1
≤ ≤
Proof. By (3.10), Lemma 3.4 and (3.9),
Ω (G) Ω (G,<) Ω (G,<) = Ω (G) 2Ω (G).
1 1 2 2 1
≤ ≤ ≤
(Alternatively, one can use a simplified version of the proof of Lemma 3.4.)
(cid:3)
Using a symmetrized version of Ω , or otherwise, it is easy to prove the
0
corresponding result for Ω .
0
Remark 3.6. If G is regular, then any order < satisfies the condition of
Lemma 3.4 and Corollary 3.5, so these results show that Ω (G,<) is the
2
same for all orders, and Ω (G,<) is the same for all orders within a factor
1
of 2; the latter holds also for Ω .
0
The factor 2 in Corollary 3.5 is annoying but not really harmful for our
purposes. It is best possible, as shown by the following example.
Example 3.7. Consider a balanced complete bipartite graph G = K
m,m
(so n = 2m), with bipartition (V ,V ). Given an order on the vertex set
1 2
≺
V V , let N := (x,y) V V :x y . Note that
1 2 ij i j
∪ ∈ × ≺
(cid:12)(cid:8) N12+N21 = V1 V2(cid:9)(cid:12)= m2. (3.13)
(cid:12) × (cid:12)
Let A V = V1 V2 and let ai =(cid:12)A Vi (cid:12), i = 1,2. Then e(v,A) = a2 if
⊆ ∪ (cid:12)| ∩ |(cid:12)
v V and e(v,A) = a if v V . Hence,
1 1 2
∈ ∈
n3Ω (G, ,A) = e(v,A) e(w,A)
1 ≺ − +
v≺w
X(cid:0) (cid:1)
= N (a a ) +N (a a ) . (3.14)
12 2 1 + 21 1 2 +
− −