Kneser ranks of random graphs and minimum difference representations

0.41 MB·

by  Zoltán Füredi

Kneser ranks of random graphs and
minimum difference representations
7
1
0 Zolt´an Fu¨redi1∗ and Ida Kantor2†
2

n
1 Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary
a
J (e-mail: [email protected])

8 2 Charles University, Prague
2
(e-mail: [email protected])

]
O
C
.
h
Abstract
t
a
m Every graph G=(V,E) is an induced subgraph of some Kneser graph of rank k, i.e.,
[ thereisanassignmentof(distinct)k-setsv A totheverticesv V suchthatA and
v u
(cid:55)→ ∈
A are disjoint if and only if uv E. The smallest such k is called the Kneser rank of G
1 v ∈
and denoted by f (G). As an application of a result of Frieze and Reed concerning
v Kneser
2 the clique cover number of random graphs we show that for constant 0 < p < 1 there
9 exist constants c =c (p)>0, i=1,2 such that with high probability
i i
2
8 c n/(logn)<f (G)<c n/(logn).
1 Kneser 2
0
.
1 We apply this for other graph representations defined by Boros, Gurvich and Meshulam.
0 A k-min-difference representation of a graph G is an assignment of a set A to each
i
7 vertex i V(G) such that
1 ∈
:
v ij E(G) min A A , A A k.
i j j i
∈ ⇔ {| \ | | \ |}≥
i
X
The smallest k such that there exists a k-min-difference representation of G is denoted
r
a by fmin(G). Balogh and Prince proved in 2009 that for every k there is a graph G with
f (G) k. We prove that there are constants c(cid:48)(cid:48),c(cid:48)(cid:48) > 0 such that c(cid:48)(cid:48)n/(logn) <
min ≥ 1 2 1
f (G)<c(cid:48)(cid:48)n/(logn) holds for almost all bipartite graphs G on n+n vertices.
min 2
0Keywords and Phrases: random graphs, Kneser graphs, clique covers, intersection graphs.
2010 Mathematics Subject Classification: 05C62, 05C80. [Furedi_Kantor_final.tex]
Submitted to ??? Printed on January 31, 2017
∗Research was supported in part by grant (no. K116769) from the National Research, Development and
Innovation Office NKFIH, by the Simons Foundation Collaboration Grant #317487, and by the European
Research Council Advanced Investigators Grant 267195.
†Supported by project 16-01602Y of the Czech Science Foundation (GACR).
1
Z. Fu¨redi and I. Kantor: Kneser ranks and min-difference representations 2
1 Kneser representations
A representation of a graph G is an assignment of mathematical objects of a given kind
(intervals, disks in the plane, finite sets, vectors, etc.) to the vertices of G in such a way
that two vertices are adjacent if and only if the corresponding sets satisfy a certain condition
(intervals intersect, vectors have different entries in each coordinate, etc.). Representations
of various kinds have been studied extensively, see, e.g., [7], [10], the monograph [15], or
from information theory point of view [13]. The representations considered in this paper are
assignments v A to the vertices v V of a graph G = (V,E) such that the A ’s are
v v
(cid:55)→ ∈
(finite) sets satisfying certain relations.
The Kneser graph Kn(s,k) (for positive integers s 2k) is a graph whose vertices are
≥
all the k-subsets of the set [s] := 1,2,...,s , and whose edges connect two sets if they are
{ }
disjoint. An assignment (A ,...,A ) for a graph G = (V,E) (where V = [n]) is called a
1 n
Kneser representation of rank k if each A has size k, the sets are distinct, and A and A
i u v
are disjoint if and only if uv E.
∈
Every graph on n vertices with minimum degree δ < n 2 has a Kneser representation of
−
rank (n 1 δ). To see that, define the co-star representation (A(cid:48),...,A(cid:48) ) of G. For every
1 n
− −
i V(G), let A(cid:48) be the set of the edges adjacent to i in the complement of G (this is the
∈ i
graph G with V(G) = V(G) and E(G) = V(G) E(G)). We have A(cid:48) A(cid:48) = 1 if ij E(G),
2 \ i∩ j (cid:54)∈
otherwise A(cid:48) A(cid:48) = 0, and the maximum size of A(cid:48) is n 1 δ(G). To turn the co-star
i ∩ j (cid:0) (cid:1) i − −
representation into a Kneser representation add pairwise disjoint sets of labels to the sets
A(cid:48),...,A(cid:48) to increase their cardinality to exactly n 1 δ(G). The resulting sets A ,...,A
1 n 1 n
− −
are all distinct, they have the same intersection properties as A(cid:48),...,A(cid:48) , and form a Kneser
1 n
representation of G of rank n 1 δ(G).
− −
(n)
Let (n) denote the set of 2 2 (labelled) graphs on [n] and let (n,k,Kneser) denote
G G
the family of graphs on [n] having a Kneser representation of rank k. G (n,k,Kneser)
∈ G
is equivalent to the fact that G is an induced subgraph of some Kneser graph Kn(s,k). We
have
(n,1,Kneser) (n,2,Kneser) (n,n 1,Kneser) = (n).
G ⊆ G ⊆ ··· ⊆ G − G
Let f (G) denote the smallest k such that G has a Kneser representation of rank k. We
Kneser
have seen that f (G) n δ. We show that there are better bounds for almost all
Kneser
≤ −
graphs.
Theorem 1. There exist constants c > c > 0 such that for G (n) with high probability
2 1
∈ G
n n
c < f (G) < c .
1 Kneser 2
logn logn
We will prove a stronger version as Corollary 12.
Z. Fu¨redi and I. Kantor: Kneser ranks and min-difference representations 3
2 Minimum difference representations
In difference representations, generally speaking, vertices are adjacent if the represent-
ing sets are sufficiently different. As an example consider Kneser graphs, where the vertices
are adjacent if and only if the representing sets are disjoint. There are other type of repre-
sentations where one joins sets close to each other, e.g., t-intersection representations were
investigated by M. Chung and West [6] for dense graphs and Eaton and R¨odl [7] for sparse
graphs. But these are usually lead to different type of problems, one cannot simply consider
the complement of the graph.
This paper is mostly focused on k-min-difference representations (and its relatives), de-
fined by Boros, Gurvich and Meshulam in [5] as follows.
Definition 2. Let G be a graph on the vertices [n] = 1,...,n . A k-min-difference repre-
{ }
sentation (A ,...,A ) of G is an assignment of a set A to each vertex i V(G) so that
1 n i
∈
ij E(G) min A A , A A k.
i j j i
∈ ⇔ {| \ | | \ |} ≥
Let (n,k,min) be the set of graphs with V(G) = [n] that have a k-min-difference represen-
G
tation. The smallest k such that G (n,k,min) is denoted by f (G).
min
∈ G
The co-star representation (which was investigated by Erd˝os, Goodman, and P´osa [8] in
their classical work on clique decompositions) shows that f (G) exists and it is at most
min
n 1 δ(G).
− −
Boros, Collado, Gurvits, and Kelmans [4] showed that many n-vertex graphs, including
all trees, cycles, and line graphs, the complements of the above, and P -free graphs, belong
4
to (n,2,min). They did not find any graph with f (G) 3. Boros, Gurvitch and
min
G ≥
Meshulam [5] asked whether the value of f over all graphs is bounded by a constant.
min
This question was answered in the negative by Balogh and Prince [3], who proved that for
every k there is an n such that whenever n > n , then for a graph G on n vertices we have
0 0
f (G) k with high probability. Their proof used a highly non-trivial Ramsey-type result
min
≥
due to Balogh and Bollob´as [2], so their bound on n is a tower function of k.
0
Our main result is a significant improvement of the Balogh-Prince result. Let (n,n)
denote the family of 2n2 bipartite graphs G with partite sets V and V , V = V =Gn.
1 2 1 2
| | | |
Theorem 3. There is a constant c > 0 such that for almost all bipartite graphs G (n,n)
∈ G
one has f (G) cn/(logn).
min
≥
Let H be a graph on logn vertices with f (H) clogn/(loglogn). One of the basic
min
≥
facts about random graphs is that almost all graphs on n vertices contain H as an induced
subgraph. ThefollowingtheoremisaneasyconsequenceofthisfacttogetherwithTheorem3.
Corollary 4. There is a constant c > 0 such that almost all graphs G on n vertices satisfy
clogn
f (G) .
min
≥ loglogn
Z. Fu¨redi and I. Kantor: Kneser ranks and min-difference representations 4
3 On the number of graphs with k–min-dif representations
3.1 The structure of min-dif representations of bipartite graphs
Analogously to previous notation, (n,k,min) (and (n,n,k,min)) denotes the family
G G
of (bipartite) graphs G with n labelled vertices V (partite sets V and V , V = V = n,
1 2 1 2
| | | |
respectively)withf (G) k. OuraiminthisSectionistoshowthatthereexistsaconstant
min
c > 0 such that (n,n,k,m≤in) = o(2n2) if k < cn/(logn). This implies that for almost all
|G |
bipartite graphs on n+n vertices f (G) cn/(logn).
min
≥
A k-min-difference representation (A : i V) of G is reduced if deleting any element x
i
∈
from all sets that contain it yields a representation of a graph different from G. Note that
A A 1 (A x) (A x) A A
i j i j i j
| \ |− ≤ | \ \ \ | ≤ | \ |
so the graph G(cid:48) corresponding to the k-representation (A x : i V) has no more edges than
i
\ ∈
G, E(G(cid:48)) E(G). There is a natural partition of the elements of A : for every = I [n],
i
⊆ ∅ (cid:54) ⊆
we have the subset ( A ) ( A ) where A is the complement of the set A . We call
i∈I i ∩ j(cid:54)∈I j j (cid:83) j
these subsets atoms. If a k-min-difference representation is reduced, then no atom has more
(cid:84) (cid:84)
thankelements. Itfollowsthatthegroundset A ofareducedrepresentationofann-vertex
i
graph has no more than k2n elements. Lemma 5 improves on this observation.
(cid:83)
Lemma 5. Let G be a graph with n vertices and (A ,...,A ) a reduced k-min-difference
1 n
representation of G. Then
A 2e(G)k kn2.
i
≤ ≤
(cid:12)(cid:91) (cid:12)
(cid:12) (cid:12)
Proof. Define the sets A := A(cid:12) A (cid:12)in the cases ij E(G), and A A = k. Let
i,j i j i j
\ ∈ | \ |
S := A . The number of elements in S is bounded above by the quantity E(G) 2k. We
i,j
| |·
claimthatS = A . Otherwise, ifthereisanelementx ( A ) S, thentherepresentation
i i
can b(cid:83)e reduced, (A x : i V) defines the same graph a∈s (A : i\ V). (cid:50)
i i
(cid:83) \ ∈ (cid:83) ∈
The upper bound in Lemma 5 can be significantly improved for bipartite graphs.
Lemma 6. Let G (n,n) be a bipartite graph with n + n labeled vertices, G
∈ G ∈
(n,n,k,min). Let (A ,...,A ) and (B ,...,B ) be the sets representing the two parts.
1 n 1 n
G
If (A ,...,A ,B ,...,B ) is a reduced k-min-difference representation of G, then
1 n 1 n
A B 4kn.
i i
∪ ≤
(cid:12)(cid:16)(cid:91) (cid:17) (cid:16)(cid:91) (cid:17)(cid:12)
(cid:12) (cid:12)
Proof. Suppose that A (cid:12) A and B (cid:12) B . Let A := A and B := B ,
1 n 1 n i i
| | ≤ ··· ≤ | | | | ≤ ··· ≤ | |
S := A B. Define
∪ n−1 (cid:83) (cid:83)
A(cid:48) := (A A ). (1)
i i+1
\
i=1
(cid:91)
Z. Fu¨redi and I. Kantor: Kneser ranks and min-difference representations 5
A
1
A
2
... ...
A
i
A
i+1
... ...
A
n
A’
Figure 1: A A in a min-dif representation when 1,2,...,n is independent
1 n
| | ≤ ··· ≤ | | { }
For each i, the inequality A A A A follows from the assumption that A
i i+1 i+1 i i
| \ | ≤ | \ | | | ≤
A . The vertices in each part of G form an independent set, so for each i, we have
i+1
| |
A A k 1. Hence A(cid:48) (n 1)k.
i i+1
| \ | ≤ − | | ≤ −
If x A A for some α < β, then there is an index i such that x A A and
α β i i+1
∈ \ ∈ \
therefore x A(cid:48). In other words, if x A A(cid:48) and α < β then x A . Therefore the sets
α β
∈ ∈ \ ∈
A A(cid:48) form a chain (see Figure 1),
i
\
A A(cid:48) A A(cid:48) A A(cid:48).
1 2 n
\ ⊆ \ ⊆ ··· ⊆ \
Treat the other part of G analogously: define B(cid:48) and note the same bound on its size, and
note that the sets B B(cid:48) form a chain.
i
\
Let us define D = S (A(cid:48) B(cid:48)). We will prove that there are at most 2(n+1) sets of the
\ ∪
form A B and B A , each of cardinality k, covering D. Therefore D contains at most
m (cid:96) p q
\ \
2(n+1)k elements. For each 1 i n, let us define A = A D and B = B D. Let
i i i i
(cid:93) (cid:93) ≤ ≤ (cid:93)∩ ∩
A = B = and A = B = D. The sets A ,A ,...,A ,A form a chain, same for
0 0 n+1 n+1 0 1 n n+1
B ,B ...,B∅ ,B(cid:93). TheelementsofD belongto(n+1(cid:102))2 atoms(asdefine(cid:102)dinthebeginning
0 1 n n+1
o(cid:102)f this(cid:102)section), many of them possibly empty, co(cid:102)rre(cid:102)spondin(cid:102)g to the squares in Figure 2.
(cid:102) (cid:102) (cid:102) (cid:93)
Foreachi,j, 1 i,j n+1, (i,j) = (n+1,n+1), theatomS isdefinedas(A A )
i,j i i−1
(cid:93) ≤ ≤ (cid:54) \ ∩
(B B ). Since the representation is reduced, no elements from the atom S can be left
j j−1 i,j
\
out, so e.g., S = . It follows that either there are some m and (cid:96) such that A(cid:102) B = k
1,1 m (cid:96)
∅ | \ |
a(cid:102)nd the atom S belongs in A B (here n m i 1 and j > (cid:96) 1), or there are some
i,j m (cid:96)
\ ≥ ≥ ≥ ≥
p,q such that B A = k and the atom S is in B A . Since A B A B , we
p q i,j p q m (cid:96) m (cid:96)
| \ | \ | \ | ⊆ \
have A B k in the first case. Likewise in the second case, B A k. In Figure 2,
m (cid:96) p q
| \ | ≤ | \ | ≤
the first option corresponds to a rectangle containing the S cell and(cid:103)the u(cid:102)pper-right corner,
i,j
witha(cid:103)llthe(cid:102)squaresinthisrectangletogethercontainingonlyatmo(cid:102)stk e(cid:102)lements. Thesecond
Z. Fu¨redi and I. Kantor: Kneser ranks and min-difference representations 6
B B B
1 2 3
f f f
A
1
f
A
2
f
A B
m ‘
\
A S
3 ij
g f
f
B A
p q
\
f f
Figure 2: The elements of S (A(cid:48) B(cid:48)) split into (n+1)2 atoms.
\ ∪
option corresponds to a similar rectangle with only at most k elements in it, containing the
S square and the lower-left corner.
i,j
Call a subrectangle A B critical if A B = k, and similarly B A is critical
m (cid:96) m (cid:96) p q
\ | \ | \
if B A = k. Our argument above can be reformulated that every (nonempty) cell S
p q i,j
| \ |
is covered by a critical r(cid:103)ectan(cid:102)gle. This implies that in each row one can(cid:102)find(cid:102)at most two
critical rectangles that cover all non-empty atoms in it. This yields the desired upper bound
D 2(n+1)k.
| | ≤
Finally, altogether S A(cid:48) + B(cid:48) + D 4kn. (cid:50)
| | ≤ | | | | | | ≤
3.2 Counting reduced matrices
Let S be a set of size S = 4kn. In this subsection we give an upper bound for the
| |
number of sequences (A ,...,A ) of subsets of S satisfying the following two properties
1 n
(P1) A A ,
1 n
| | ≤ ··· ≤ | |
(P2) A A k 1 (for all 1 i n 1).
i i+1
| \ | ≤ − ≤ ≤ −
Let be the 0-1 matrix that has the characteristic vectors of the sets A ,...,A as
1 n
M
its rows (in this order). The positions in where an entry 1 is directly above an entry 0
M
will be called one-zero configurations, while the positions where a 0 is directly above a 1 will
be called zero-one configurations. A column in a 0-1 matrix is uniquely determined by the
locations of the one-zero configurations and the zero-one configurations unless it is a full 0
or full 1 column. We count the number of possible matrices by filling up the n (4kn)
M ×
entries in three steps.
Z. Fu¨redi and I. Kantor: Kneser ranks and min-difference representations 7
Each one-zero configuration corresponds to an i < n and to an element x A A . A
i i+1
∈ \
set A A can be selected in at most
i+1 i
\
4nk 4nk
+ + < (4en)k
0 ··· k 1
(cid:18) (cid:19) (cid:18) − (cid:19)
ways (n > k 1). Do this for each i < n, altogether we have less than (4en)kn ways to write
≥
in the one-zero configurations into .
M
Select in each column the top 1. If there is no such element in a column we indicate
that it is blank, a full zero column. There are at most n + 1 outcomes for each column,
altogether there are at most (n+1)4kn possibilities. Fill up with 0’s each column above its
top 1. Define A(cid:48) S as in (1), A(cid:48) := n−1(A A ). We have A(cid:48) kn. The columns of
⊂ i=1 i \ i+1 | | ≤
that correspond to the elements of S A(cid:48) have a (possibly empty) string of zeros followed
M (cid:83)\
by a string of ones. We almost filled up and we can finish this process by selecting the
M
remaining zero-one configurations.
There may be several zero-one configurations in a single column. Each of them has a
unique (closest, or smallest indexed) 1 above them. That element 1 is already written in into
our still partially filled , because that element 1 (even if it is the top 1 element) belongs to
M
a unique one-zero configuration. This correspondence is an injection. So there are at most
A A kn zero-one configurations in the columns corresponding to A(cid:48) which are
i| i \ i+1| ≤
not yet identified. There are at most nkn ways to select them.
(cid:80)
Since (for n > k 1)
≥
(4en)kn (n+1)4kn nkn < n6kn+O(kn/logn) = e6knlogn+O(kn),
× ×
we obtain the following
Claim 7. Altogether, there are eO(knlogn) ways to fill with entries in 0,1 according to
M { }
the rules (P1) and (P2).
3.3 Proofs of the lower bounds
Proof of the lower bound in Theorem 3. Let G = G(V ,V ) be a bipartite graph with both
1 2
parts of size n and suppose that G belongs to (n,n,k,min). By Lemma 6 we may suppose
G
that G has a reduced k-min-difference representation (A ,...,A ,B ,...,B ) such that each
1 n 1 n
representingsetisasubsetofS,where S = 4kn. Therearepermutationsπandρ S which
n
| | ∈
rearrange the sets according their sizes A A and B B .
π(1) π(n) ρ(1) ρ(n)
| | ≤ ··· ≤ | | | | ≤ ··· ≤ | |
Consider the V S matrices, , i = 1,2 whose i’th row is the 0-1 characteristic vector of
i i
× M
A and B , respectively. The permutations π, ρ and the matrices , completely
π(i) ρ(i) 1 2
M M
describe G. The matrices satisfies properties (P1) and (P2), so Claim 7 yields the
i
M
following upper bound for the number of such fourtuples (π,ρ, , )
1 2
M M
(n,n,k,min) #of(π,ρ, , )(cid:48)s (n!)2n(12+o(1))kn = eO(knlogn). (2)
1 2
|G | ≤ M M ≤
Z. Fu¨redi and I. Kantor: Kneser ranks and min-difference representations 8
Here the right hand side is o(2n2) if k 0.057n/(logn) implying that f (G) >
min
0.057n/(logn) for almost all the 2n2 bipartite≤graphs. (cid:50)
Proof of the lower bound for the random bipartite graph.
Recall that in a random graph G (n,p), each of the n edges occurs independently with
∈ G 2
probability p. Similarly, (n,n,p) denotes the class of graphs (n,n) with the probability of
G (cid:0) (cid:1) G
a given graph G (n,n) is
∈ G
pe(G)(1 p)n2−e(G).
−
Here the right hand side is at most (max p,1 p )n2. This implies that for any class of
{ − }
graphs (n,n) the probability Pr(G ) is at most times this upper bound. If the
A ⊂ G ∈ A |A|
class of graphs is too small, namely
A
1 1
= o (min , )n2 ,
|A| {p 1 p}
(cid:18) − (cid:19)
then for (n,n,p) one has
G
Pr(G ) 0. (3)
∈ A →
Taking := (n,n,k,min) with a sufficiently small k, we obtain
A G
Corollary 8. For constant 0 < p < 1 there exists a constant c = c (p) > 0 such that the
1,min
following holds for G (n,n,p) with high probability as n
∈ G → ∞
n
c < f (G).
min
logn
4 Maximum and average difference representations
Boros, Gurvich and Meshulam [5] also defined k-max-difference representations and k-
average-difference representations of a graph G in a natural way, that is, the vertices i and j
areadjacentifandonlyifforthecorrespondingsetsA ,A wehavemax A A , A A
i j i j j i
{| \ | | \ |} ≥
k and ( A A + A A )/2 k, respectively. Analogously to f we can define f (G)
i j j i min max
| \ | | \ | ≥
andf (G). Sincefor every graphGa Kneserrepresentation is amin-dif, average-difference,
avg
and max-difference representation as well we get
f (G), f (G), f (G) f (G) n 1. (4)
min avg max Kneser
≤ ≤ −
Let (n,k,max), (n,n,k,max), ( (n,k,avg), (n,n,k,avg)) denote the family of graphs
G G G G
G (n) and in (n,n) with n labeled vertices V or with partite sets V and V , V =
1 2 1
∈ G G | |
V = n, respectively, such that f (G) k (f (G) k, respectively).
2 max avg
| | ≤ ≤
It was proved in [5] that f and f are not bounded by a constant, for a matching of
max min
size t one has f (tK ) = Θ(logt) and f (tK ) = Θ(logt). (It turns out that f (tK ) =
max 2 avg 2 min 2
1.) The proof of Theorem 3 can be easily adapted for these parameters for (n,n,p) as well.
G
Even more, we can handle the general case G (n,p), too.
∈ G
Z. Fu¨redi and I. Kantor: Kneser ranks and min-difference representations 9
Corollary 9. For constant 0 < p < 1 there exists a constant c = c(p) > 0 such that the
following holds for G (n,n,p) with high probability as n
∈ G → ∞
n
c < f (G), f (G). (5)
avg max
logn
Similarly for G (n,p) with high probability we have
∈ G
n
c < f (G), f (G). (6)
avg max
logn
These lower bounds together with the upper bounds from Corollary 14 below imply that
for almost all n vertex graphs, and for almost all bipartite graphs on n+n vertices, f (G)
avg
and f (G) are Θ(n/(logn)).
max
Sketch of the proof. If G (n,n,k,max) (and if G (n,n,k,avg)) and (A ,...,A ,B ,
1 n 1
∈ G ∈ G
...,B ) is a k-max-difference (k-average-difference) representation then
n
A A 2k 2 ( 2k 1) (7)
i j
| (cid:52) | ≤ − ≤ −
holds for each pair i,j. If the representation is reduced, then we obtain (without the tricky
proof of Lemma 6) that A < 2kn, and the same holds for B , too. The conditions
| i i| | i i|
of Claim 7 are satisfied implying
(cid:83) (cid:83)
(n,n,k,max) , (n,n,k,avg) = eO(knlogn).
|G | |G |
We complete the proof of (5) applying (3) as it was done at the end of the previous Section.
Consider a graph G (n,k,max) and let (A ,...,A ) be a reduced k-max-difference
1 n
∈ G
representation. (The case of k-average-difference representation can be handled in the same
way, and the details are left to the reader). The only additional observation we need is that
since (7) holds for each non-edge i,j , we have A A 4k 4 for all pairs of vertices
i j
{ } | (cid:52) | ≤ −
whenever diam(G) 2. Thus for every reduced representation (in case of diam(G) 2) one
≤ ≤
has A (4k 4)n. Also, A A 2k 2 for A A . Then the conditions of
| i i| ≤ − | i \ j| ≤ − | i| ≤ | j|
Claim 7 are fulfilled (with 2k 2 instead of k) implying the following version of (2)
(cid:83) −
(n,n,k,min) (n) = eO(knlogn),
2
|G \G |
where (n) denotes the class of graphs with G (n),diam(G) > 2.
2
G ∈ G
We complete the proof of (6) by applying (3) and the fact that
diam(G) 2
≥
holds with high probability for G (n,p). (cid:50)
∈ G
Z. Fu¨redi and I. Kantor: Kneser ranks and min-difference representations 10
5 Clique covers of the edge sets of graphs
We need the following version of Chernoff’s inequality (see, e.g., [1]). Let Y ,...,Y be
1 n
mutually independent random variables with E[Y ] 0 and all Y 1. Let a 0. Then
i i
≤ | | ≤ ≥
Pr[Y + +Y > a] < e−a2/(2n). (8)
1 n
···
A finite linear space is a pair (P, ) consisting of a set P of elements (called points) and
L
a set of subsets of P (called lines) satisfying the following two properties.
L
(L1) Any two distinct points x,y P belong to exactly one line L = L(x,y) .
∈ ∈ L
(L2) Any line has at least two points.
In other words, the edge set of the complete graph K(P) has a clique decomposition into the
complete graphs K(L), L .
∈ L
Lemma 10. For every positive integer n there exists a linear space = with lines
n
L L
L ,...,L such that m = n+o(n), every edge has size (1+o(1))√n, and every point belongs
1 m
to (1+o(1))√n lines.
Proof. (Folklore). If n = q2 where q > 1 is a power of a prime then we can take the q2 +q
lines of an affine geometry AG(2,q). Each line has exactly q = √n points and each point
belongs to q+1 lines. In general, one can consider the smallest power of prime q with n q2
≤
(we have q = (1+o(1))√n) and take a random n-set P F2 and the lines defined as P L,
q
⊂ ∩
L (AG(2,q)). (cid:50)
∈ L
5.1 Thickness of clique covers
The clique cover number θ (G) of a graph G is the minimum number of cliques required
1
to cover the edges of graph G. Frieze and Reed [9] proved that for p constant, 0 < p < 1,
there exist constants c(cid:48) = c(cid:48)(p) > 0, i = 1,2 such that for G (n,p) with high probability
i i ∈ G
n2 n2
c(cid:48) < θ (G) < c(cid:48) .
1(logn)2 1 2(logn)2
They note that ‘a simple use of a martingale tail inequality shows that θ is close to its mean
1
with very high probability’. We only need the following consequence concerning the expected
value.
n2
E(θ (G)) < c(cid:48) . (9)
1 3(logn)2
The thickness θ of a clique cover := C ,...,C of G is the maximum degree of the
0 1 m
C { }
hypergraph , i.e., θ ( ) := max deg (v). The minimum thickness among the clique
0 v∈V(G) C
C C
covers of G is denoted by θ (G).
0
A clique cover corresponds to a set representation v A in a natural way A := C :
v v i
C (cid:55)→ {
v C with the property that A and A are disjoint if and only if u,v is a non-edge of G.
i u v
∈ } { }