Commentationes Mathematicae Universitatis Carolinae
Jaroslav Nešetřil; Vojtěch Rödl
Partitions of vertices
Commentationes Mathematicae Universitatis Carolinae, Vol. 17 (1976), No. 1, 85--95
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OÓMMENTATIONES MATHEMATICAE UNI7ERSITATIS CAROLINA!
17,1 (1976)
PARTITIONS 0.F VERTICES
J a r o s l a v NEŠETŘIL, Vojtěch BffDL, Praha
.DEDICATED T0 EMANUEL GRINBERG ON THE OCCASION OF HIS 65-TH
BIRTHDAY
Abstract:
In t h i s note we show how a theorem by ErdosHajnal may%e used f o r proving theorems concerned with par­
t i t i o n s of vertices of graphs, relations etc*
Key-words:
Ramsey theorem, partitions, chromatic number.
AMS: 05A99
Ref. 2 . J 8.83
Introduction.
In 1966 Erdos and Hajnal t l ] proved the
following.
Theorem A :
n>l
.For every positive integer
there e x i s t s a hypergraph
tf
k z 2 , t>2
,
» ?f(k, Z fn) « CX,7Jt)
with the following properties:
1J!
^
is a
2)
$
does not contain cycles of length smaller than
3)
ft
(<f)> n
k-uniform hypergraph
The notation i s the following:
of v?
fy
monocolcured hyperedge occurs;
ce
(^f ) - chromatic number
i . e . the minimal number of colours which are necessa­
ry for colouring the vertices of Jf
- k
i
for every
/
M e tfl
x 1 ,M J? x 2 ,M2,...,x J g > M^
in such a way that no
k-uniform means that
; a cycle of length Z
such that
- 85 -
*.,€ M^ ,
IMI -
i s a sequen­
i e C 1, 11 5
x
x
i + l * % , i € t l , .4 - 1 1 ;
^ l i t t l . - t J J S
W
, {x,
l6 %
j i € [ 1 , Z 11 £ X .
To avoid the t r i v i a l cycle consisting of only one hyperedge
we assume that there are
i, j
such that
M ^ M.* •
Theorem A was proved by nonconstructive means. In 1968 L.
Lov&sz proved
the same theorem constructively.
In t h i s note we show how t h i s theorem implies (using a simpl e t r i c k ) a very general theorem of Ramsey type for p a r t i t i o n s of v e r t i c e s . There are two reasons for publishing of
t h i s note: f i r s t , the t r i c k provides simpler proofs t o known
theorems (C23-.C3],C4l), secondly, p a r t i t i o n s of v e r t i c e s a r e
used as a t o o l for proving a Ramsey type theorem for parti—
tions of edges and we s h a l l need a general
theorem for p a r -
t i t i o n s of v e r t i c e s for our forthcoming papers.
We apply the Theorem A t o p a r t i t i o n s of v e r t i c e s of graphs,
hypergrapha, r e l a t i o n s and universal algebras. In § 4 we show
t h a t given a graph
mal
G t h e r e e x i s t s an i n f i n i t e set of m i n i -
graphs with the vertext p a r t i t i o n property for
G . We
end t h i s note with a few problems and comments concerning
i n f i n i t e graphs.
1. •ffolkman's the ore nu
For every p o s i t i v e integer
In 1967 J . Jfolkman [33 proved;
r
and for every graph
G -=- (V,E)
without complete subgraphs on m v e r t i c e s t h e r e e x i s t s a
graph
H = (W,iT)
without complete subgraphs on
such that for every p a r t i t i o n
i
and mi embedding
bedding
f: G—*>H
f: G—*H
is an
W - * w . W4
such that
m vertices
there e x i s t s a n
f(V)S W.^ (An em-
1-1 mapping with the property
- 86 -
-Cf(x),f(y)Jc -?<«=*>ix,y}c E . ) . We denote by G - 2 1 ^ B
the validity of the above statement for
i s denoted by
G --/
sense* Let
G—»H
bedding of
G into
G, H , the negation
> H • This notation has the following
denote the fact that there exists an emH . Then G
re are "so manywembeddings of
partition vertices of
H into
>
G into
r
H means that the-
H that even i f we
parts we s t i l l have an em-
bedding i n one of the parts. In t h i s way
seen as a combinatorially strengthened
—-^—*»
may be
embedding arrow (see
C83) .
.Folkman gave a direct constructive proof or the above fact.
An another ( l e s s elementary) proof i s due to the authors of
C73« However, Theorem A instantly yields a much stronger result.
Definition:
Let
K be a fixed graph. Denote by Gra (K)
the class of a l l graphs which do not contain
graph. ( I . e .
K as a sub-
G <£ Gra (K)<«s=s> there are sets
such that
If X
(Vo*
rt,E o ) S* K . )
i s a set of graphs put
VQcV , £ Q c £
Theorem 1:
graphs. Then for
Let X
Gra (X)
= C\ (Gra (K) 1 K € X ) .
be a f i n i t e set of
every graph
graph He Gra CSC) such that
GsGra (X)
2-connected
there exists a
G .———> H .
to
We may assume \K\y2
for every K e X
as for
\&[£2
we
get either the void class of graphs of the class of a l l d i s crete graphs.
Proof: Let
= max I K I + 1 ,
Ke3C
G =- (V,E) Gra (3C) be fixed. Let b ~
| G | =- k . Let us choose tf (k, Z ,r) - (X,#t)
- 87 -
with t h e p r o p e r t i e s of Theorem A. .For each
fM:
V—>M
« (X,F)
Aztt
M C 1ft
let
be a fixed b i s e c t i o n . Define t h e graph
H»
such that
{x,y}cF<a«>> there exist
} e B auch t h a t 4 % ( z ) , f M ( t ) \ = i x , y ? . This graph H
w i l l be denoted by
As
M e Tfl and
X > 2
(X,77l) # G •
we have
lMONI^l
whenever
M4»N , 1M9$} c m
(see above) and consequently
1)
fM:
V—*lff i s an embedding of
i f m
.
2)
K'
If
i s a subgraph of
H ,
G into
H for each
K ' S K C Ot then K ' Q
S (M t #) , If * 491 .
(This follows by the 2-connectivity of
that
(X,W
.Finally
K and by the fact
does not contain a cycle of length < I K I + 1 • )
G ——*- H
follows immediately from
% (Xt7H) >
>r :
Given a p a r t i t i o n
i etl,r 3
X =- - O , X4
such t h a t
ced subgraph of
there e x i s t s
M§X i « Consequently
(X^,.?)
and
Me W
and
G i s an i n d u -
fjy- i s an embedding.
This theorem does not hold for graphs with connectivity
i)
then
If
K i s disconnected and
H •/
™; » K
for every
K = K'L; K"
H€Gra (K)
where
-< 2
K'£* K*
as may be seen
easily*
ii)
If
K = Pn
i s a path of length
n
then
GcGra (P n ) «=->
-,.. •"> ^,(G)-4n • From t h i s follows t h a t t h e r e e x i s t s
eGra (P^)
such t h a t
e G r a (P n )
( i t suffices to take
fies
%(Q) > 2~
G /
'* > H for every graph
; obviously
-
GeGra (P n )
G
88 -
^
>
G 6
H e
which s a t i s -
H —» ^(H) >
Z2
^(G)• - 1 ) .
iii)
If %
i s an
false (consider
cycles).
2
*
graphs.
i n f i n i t e s e t then the statement may be
X 'MC^k+l I k - 1 ?
the
set
of
a11
od<i
P a r t i t i o n s of v e r t i c e s of r e l a t i o n s and h.vperUsing
the same ideaaa i n 1
we may prove analo-
gous theorems for r e l a t i o n s and hypergraphs. We l i s t only
statements:
Theorem 2a : Let
%
be a f i n i t e s e t of
2-weakly con-
nected r e l a t i o n s (see 152 » p.199) . Then for every p o s i t i ve integer
r
S € SteJL (%)
and for every
such t h a t
Theorem 2b:
are
R —£—>
Let if
2-connected ( i . e .
R € Slejl (31)
(Xf^Tl)
r
there exists
(Y f 7l)eHyp (5f)
(xf«i) . ^
>
2-connected <a—>
2-connected graph).
and for every
(X$ffl)& Hyp (£P)
such that
(rf-u> •
The definitions of fle£ (%)
ñГ
is
is a
Then for positive integer
->
S .
be a f i n i t e of hypergraphs which
<«=-===> (XttfXPr(M) I M C W ) )
JC
there e x i s t s
and
Hyp(Jf)
and of symbols
are quite analogous t o the definitions
Gra (3C)
and
> for graphs. Again in certain sense these theorems are
best possible
3.
Partitions of algebras.
Let V
be the class of
all finite universal algebras of given type
- 89 -
A
=
(n^ | i c I ),
Let
3e=* (X,C<«>t 1 i d ) )
from V
• We write
X » j ^ ^ r.|
:
% —> %
Theorem 3:
€ X
that
(X,i*,±
and
Let
\ i el))
be algebras
i f f for every partition
there e x i s t s an i
9£ —* % such that
€ V
, %*
and a monomorphism f %
f (*) £ X t •
r
be positive integer and
ca^x,#,... * ) « *
for every
l e t 9£ e
lei
and
xc
(idempotent algebras). Then there e x i s t s ty m V such
$£ —%-»> %
.ftp oof:
Cx',m) »
Let
tf(kf3,r)
•
|X|«k,i'«3 , n«p
. Let
y ' c l ' • -for
. Consider
every
us choose a bisection
fMz X—--*>M . Define
by X « x ' u * y ' j
^ ( y j ) j J € tltn±l
and
« % ( ^ i ^ 4 1 4 e C l , * ^ ] ))
where
M e x i s t s , otherwise we put
Mem
let
<Y,C ae^ | i e X ) >
) «
%(-Cj) * J*
i f such an
1
^ ( y ^ | i « CI* -* J ) * y ' , i € I
I t i s easy to check that ty, c 1f 7
% —• > OL .
Again i t i s easy to see that, generally, for non-idempoten*
algebras Theorem 3 f a i l s to be true.
Remark:
A very d i f f i c u l t problem seems to be the cha-
racterization of those primitive classes of algebras for
which the statement analogous to Theorem 3 holds. This i s
true for example tor the class of a l l f i n i t e distributive
lattices*
4*
say that
Cr:ttl<?»A f9l,kmTi KfiPteH i s an irreducible
G —-—» H
but
I*t
G be a graph. We
(r,v)~graph for
G •/ I > H
- 90 -
for every
G if
proper subgraph
H of
H.
*
Theorem 4 a;
For every graph
G , I G ) > 1 there ex­
i s t * a countable s e t of non-isomorphic irreducible
graphs for
Proof:
(r,v)~
G •
A proof folio*© d i r e c t l y from the *c
t i o n in !• Let
G be fixed* We may assume that
construc­
G i s a con­
nected graph (otherwise we consider the complement of
I t suffices t o put
G) •
% * <f ( 1G I , 3 , r ) * G
%. «
HQ^
Sf ( I G I , I Ex | ,r) # G
» tf ( ІGІ, | H n l , l O * G
6
** -» H^ holds for every
le
(r,v)-graph for
Obviously
Assume
i . Let
G contained in
IHJJ < l H4I for a l l
H^fi-Hj
for
-* tf ( I G I , I H j ^ I ,r) #
I E± 1 S «? * G where
?
f
H^ ,
i, j
i < j . As
H^ be an irreducib­
satisfying
H^SHj ,
and
i = 1,2,... •
i< J •
Hj «
I H ^ I fe | E± \ we have
£ & ( I G | , I H j ^ I , r)
is a
hypergraph which does not contain any cycle. But i n t h i s
case
% C W * G) »
^ (G) • This contradicts
G - - J - > Ht .
Remark 1:
Using a modified proof we may even prove
- 91 -
Theorem 4 b :
#
Let %
be a f i n i t e s e t of
graphs. Then for every graph
GeGra (X)
2-connected
there exists a
countable set i R± i - 1 , 2 , . . . } of non-isomorphic graphs
such t h a t
1>
H^Gra
2)
E±
(X)
i s an i r r e d u c i b l e
Remark 2 :
(r f v)-graph f o r
G .
Theorem 4a does not hold f o r i n f i n i t e graphs.
Every complete graph of i n f i n i t e c a r d i n a l i t y i s the only
( r f v ) - i r r e d u c i b l e graph for i t s e l f . Theorem 4a f a i l s to be
true for
I G | =- 1
lemark 3 :
write
G - it
f
Let
too •
G « (V,E)
f
H • (w,*1)
> H i f for every p a r t i t i o n
r e e x i a t s an embedding
I <xfy} 6 I } S i^
be graphs. We
F » . ^
F*
the-
£: ^—^H such t h a t - C l f ( x ) f f ( y ) J )
for an
i e [ l f r 1 • The existence of an
Ramsey graph for every f i n i t e graph was proved independently
by Beuber, Erdoa, Hajnal, Posa and Rodl C93, see also much
atronger C73.
Define
G
of
H t o be an
^ - > H but
( r , e ) - i r r e d u c i b l e graph for
G /
G if
< » H ' for a l l proper subgraphs
H .
Problem 1:
Characterize those f i n i t e graphs
G for
which there e x i s t s an i n f i n i t e set of non-isomorphic
irredueible graphs for
If
(rfe)-
G•
a graph G contains at most one edge then there e x i s t s
precisely one
G
H#
,e » H
f
( r , e )-irreducible graph
namely
G itself.
92 -
H such that
Conjecture:
For a f i n i t e graph
G the following two
statements are equivalent:
!•
For G t h e r e e x i s t s a countable set of
(r, © )-irredu-
cible graphs.
2.
G / * > G .
The path of lexigth
2
i s an example of a graph
t h e r e e x i s t s a countable s e t of
G for which
( 2 , e ) - i r r e d u c i b l e graphs.
One can take the family of a l l odd cycles.
More generally, the same i s true for every paths of length
Z
9
& finite.
F i n a l l y l e t us remark that Theorem 1 shows the power of ErdosHajnal theorem for p a r t i t i o n s of v e r t i c e s .
There i s no general method known for deriving similar theorems
for p a r t i t i o n s of edges (see C8] for r e s u l t s in t h i s d i r e c t i o n ) . Let us add a few remarks concerning i n f i n i t e graphs. In
an obvious way we may extend the symbol G -—-—> H for i n f i n i t e graphs
G , H and
any cardinal
r . The following i s
then t r u e :
Theorem 5a :
teger
r
there e x i s t s
Theorem 5bs
nal
r
For every graph
H such that
G and every positive i n G —-—> H •
For every f i n i t e graph
there e x i s t s a graph
H such that
G and every cardiG - ^ ••> H • More-
over, i f
G does not contain a complete graph on m v e r t i -
ces then
H may be chosen with the same property.
Theorem 5a may be proved by the following construction:
Let
G = (V,E) , assume without loss of generality
- 93 -
r -» 2
(this is possible as G — ~ - ^ H - ** » I = s G
* x' {y»y'J € E
Given
jr I ).
where <(x,y),(x%y#)}€Jl<-«5> either x «
H = (VxY,.?)
Put
^
or «{x,x'j € E •
a colouring
c: V x ? — • t l f 2 ?
xcT
such that
0(4x3x7) =- i
that
for every
x6?
either there e x i s t s
or there e x i s t s
there e x i s t s y
ir
From t h i s follows e a s i l y G v » H .
with
i
such
cC(x,y) - i •
Theorem 5b follows from the Erdoa-Rado generalization of the
c l a s s i c a l .Ramsey theorem for cardinal numbers and from the
representation of f i n i t e graphs by type-graphs, see t73 f C83 •
This i s a straightforward application of type-graphs and we
ommit the proof*
This leads to the following problems (see also £ 2 ] ) :
Problem 2:
Let
G be a graph and r
a cardinal number.
Does there exist a graph
H such that
Moreover, providing that
G does not contain a complete graph
with
G • *f > H ?
m vertices i s i t possible to choose
H with the same
property?
Not much i s known, even the case
m= 3
and r « 2
solved. The purpose of this remark i s to show that
i s uneven
dea-
l i n g with vertex partitions one cannot be overoptimistic.
R e f e r e n c e s
[0]
[11
P.
:
Problems and results on f i n i t e and i n f i n i t e
graphs, i n : Recent advances in graph theory, Academia, Prague (1975), pp.183-192.
ERJDffS P . , HAJNAL A.: On chromatic number of graphs and
set systems. Acta Math* Acad. Sci. Hungar, 17
ERDCS:
(1966), 61-99.
-94--
[23
ERD5S P . , HAJHAL A # : Problems and r e s u l t s i n f i n i t e and
i n f i n i t e combinatorial a n a l y s i s , Ann. of N.Y. Acad.
S c i . , 175(1970), 115-124.
[3]
K)LKMAN J . : Graphs with monochromatic complete subgraphs
i n every edge colouring, SIAM J . Appl. Math. 18
(1970), 19-29.
[41
R.L. GRAHAM, B.L. ROTHSCHILD: Some recent developments
in Ramsey theory, i n : Mathematical centre t r a c t s ,
Amsterdam, 57C1974), pp.61-67.
C5]
HABART F . : Graph theory, Addison-Wealey, Reading (1969).
t6]
LOVlsz L . : On chromatic number of f i n i t e set-systema,
Acta Math. Acad. S c i . Hungar. 19(1968), 59-67*
[?]
NE§ET$IL J . , B5DL V.: The Ramsey property for graph®
with forbidden complete subgraphs, to appear in
Journal of Comb. Theory.
[ 8]
J#
V. B5DL: Type theory of p a r t i t i o n problems
of graphs, i n : Recent advances in graph theory,
Academia, Prague (1975), 405-412.
T93
PROCEEDINGS of Set theory conference, K£szth£ly (1973).
NESET&IL,
[ 1 0 ] B5DL V.: The dimension of a graph and a generalized Ramsey theorem, Thesis Charles University Praha
(1973).
Matematicko-fyzik&lni fakulta
.Fakulta jadernd fyziky a
Karlova universita
inSen^ratvf fiVUT
Sokolovsk6 83, 18600 Praha 8
Husova
Geskoslovensko
5, Praha 1
fieakoslovensko
(Oblatum 6.11.1975)
- 95 -