On Almost Bipartite Large Chromatic Graphs

On Almost Bipartite Large Chromatic Graphs

Annals of Discrete Mathematics 12 (1982) 117-123 @ North-Holland Publishing Company ON ALMOST BIPARTITE LARGE CHROMATIC GRAPHS P. ERDOS, A. HAJNAL ...

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Annals of Discrete Mathematics 12 (1982) 117-123

@ North-Holland Publishing Company

ON ALMOST BIPARTITE LARGE CHROMATIC GRAPHS

P. ERDOS, A. HAJNAL and E. SZEMEREDI Hungarian Academy olSciences. Budapest. Hungary

Dedicated to Professor A. Kotzig on the occasion of his sixtieth birthday 0. Introduction

In the past we have published quite a few papers on chromatic numbers of graphs (finite or infinite), we give a list of those which are relevant to our present subject in the references. In this paper we will mainly deal with problems of the following type: Assuming that the chromatic number ~ ( 3of)a graph $ is greater than K , a finite or infinite cardinal, what can be said about the behaviour of the set of all finite subgraphs of $. We will investigate this problem in case some other restrictions are imposed on 3 as well. Most of the problems seem difficult and our results will give just some orientation. The results show that ~ ( 4 8 can ) be arbitrarily large while the finite subgraphs are very close to bipartite graphs. It is clear from what was said above that this topic is a strange mixture of finite combinatorics and set theory and we recommend it only for those who are interested in both subjects. Finally we want to remind the reader the most striking difference between large chromatic finite and infinite graphs which was discovered by the first two authors about fifteen years ago [4]. While for any k < o there are finite graphs with z(Y)>k without any short circuits, [ l ] , a graph with x ( ~ ) > K ~ ohas to contain a complete bipartite graph [k, K ' ] for all k < w. Hence such a graph contains all finite bipartite graphs, though it may avoid short odd circuits. Our set-theoretical notation will be standard as for graph theory we will use the notation of our joint paper with F. Galvin [3] with some self-explanatory changes.

1. The standard examples and the ordered edge graph

Definition 1.1. For each ordinal a, for 2 < k < o,and for 1< i
is the graph with V#(a,k, i)= Calk, where E(a, k, i ) is defined by the following stipulations. For X E Calk we write X = { x o , . . . , xk- 1 ) with xo< . . . < x k - and for X , Y€[aIk {X, Y } EE(a,k,i) iff y j = x i + j f o r j < k - i - l . We call Y0(a,k, 1)= %(,a,

k) and we call this graph the k-edge graph on a. 117

P . Erdos et al.

I18

Definition 1.2. For each ordinal a, for 3 < k < o and for 1 < i < k - 1, Y1(a, k, i)= (Vl(a, k, i), &(a, k, i ) )

is the graph with Vl(a,k, i)= Calk where for X, Y E [.Ik

x, YEEl(a, k , i )

iff

xi

We call Yl(a, k , i ) the k, i-Specker graph on a, and just as in the case above we omit i for i= 1. The above graphs are standard examples of large chromatic graphs, we list some properties of them.

Lemma 1.1. (For the proofs see [3] and [4]). (a) x(Yo(a, k , i ) ) > K prooided a-(2k)f: and, as a corollary of this, for all x(Y0(@,k, i ) ) > K prooided

K),o,

a2(expk-,(ic))+ f o r a l l 2 < k < o , l
+ 00

X(Yl(n,k, i))-

if n+

+ 00.

(c) 4B0(a,k ) does not contain odd circuits of length 2j+ 1 for 1< j < k - 1. (d) Y1(a,k ) does not contain a complete rc-graphfor 3
Definition 1.3. For a graph Y = ( K E) and for an ordering < of V we define the ordered edge graph OE(48, <)=(V(Y, <),E(Y, <))of Y for the ordering <, as follows V(48,<)=E. For X E [ V ] write ~ X = { x o , x l } where xo
yl=xo.

It is clear from the definition that the 2-edge graph Yo(a,2) is the ordered edge graph of the complete graph with vertex set a for the natural ordering of a. More generally the following is true:

Lemma 1.2. For each a, and for 1 < k < o there is a well-ordering

can be any ordering of

min X
[.Ik

satisfying

3 x
Lemma 1.3. Let Y be a graph with x(Y)<2" for the vertices. Then x(OE(Y, < ) ) < 2 ~ .

Proof. Let Y be Y =

( x E). Then

K),

1 and let

< be any ordering of

V is the union of 2" independent sets. Since the

Almost bipartite large chromatic graphs

119

complete graph of 2" vertices is the union of K bipartite graphs, there are disjoint partitions V = A i u B i of V for i < K such that E c U i < , [ A i , Bi]. We now partition each [Ai, Bi] into the union of two sets independent in OE(Y). Let Ci= { {xo,x , ~ E} [Ai, Bi]: x j E Ai} for j < 2 . Clearly Ci contains no edge of OE(Y) and CbuCi=[Ai, Bi] for i < K . In what follows log denotes the 2 basis logarithm and log'k) is the k-times iterated logarithm function.

Corollary 1.4. If Y is a subgraph of n uertices of some Yo(a,k ) for k >,2 then x(Y)
log"-"(n)

for some ck>O.

To close this section we mention the first problem we cannot solve.

Definition 1.4. For each (infinite) graph Y= ( V , E) let f&)=max{X(Y(A)): A c V ~ l A l = n } .

Problem 1. For what functions f : w + o is it true that for all cardinals is a graph 48 with x(Y)> K and f$(n)
K>O

there

Corollary 1.4 shows that 10g'~)(n)is such a function for all k w there is a graph with x(48)> K all whose finite subgraphs are in $ then 9 must contain all finite subgraphs of Yo(o, k) for some k
2. Omitting vertices of subgraphs DefinitiOn 2.1. Let Y = ( V , E) be a graph; fsl'(n)=min{max{lZI: Z c A ~ Z i s i n d e p e n d e n t ) .A: c l / ~ l A ( = n ) , f&~)=min{max{lZI: Z c A r \ Y ( Z ) i s bipartite): A c V ~ ' l A l = n ) . Clearly fi(n)>if&n) holds. We are interested in the problem if these functions can be large for a graph of large chromatic number. The following theorem contains our main information about this.

120

P. Erdos et al.

Theorem 1. For all E > 0 and for all K there is a graph Y with ~ ( 4 6>) K such that f A n ) >,( 1 - E ) n

holds for all n < w .

Proof. By Lemma 1.1(a)we only have to show that for Yk = Yo(@,k ) f:,(n) >,( 1 -2/k)n.

Let A c [0Ik,( A (= n, for some 2 < k < w and for some n. We prove, by induction on n, thqt there is a set Z c A, [Zl a(1 - 1/2k)n such that Z spans a bipartite graph of Y(o, k ) . The statement is trivial for n = 1. Assume n > 1 and that the statement is true for all n k n . ForxEWlet S ( X ) = { X E A x : ~ X } , a n dP ( x ) = { X ~ A : x = x ~ f ol O and IP(x)l>,(1-2/k)lS(x)J. Now the graph induced by P(x) can be shown to be bipartite bythefollowingpartition:Ao={X ~ P ( x ) : x = x ~ ~ i i s e v e n ) , (AX, ~= P ( x ) : x = x ~ r \ i is odd}. A o u A l =P(x) and A j contains no edge of Y(w, k) for j < 2 . By induction, there is a subset Q c A\S(x), IQI >,( 1 - 2/k)(n - (S(x)J),which induces a bipartite graph of Yo(o, k). Since no edge of Yo(w, k) joins a point of P(x) to a point of A W x ) , Z = P(x)uQ satisfies the requirements of the theorem.

,

The theorem yields that there is a sequence E,+O so that for some Y with x(Y)=w, n( 1 - E J . We do not know how fast E, can tend to 0. A result of Folkman gives information for the case $ n - f d ( n ) is bounded. This says that if i n - f J ( n ) < k then x(Y) < 2k 2. Here we at least know that the situation is different for graphs with chomatic number >o. &I)>

+

Lemma 2.1. !fx(Y)>w then there is an E > O such that f$(n)<($-E)n.

hf. Clearly Y contains HIvertex disjoint copies of some C Z i + for , some i < w . The union of m copies has cardinality rn(2i+ I ) and contains no free set larger than mi. Problem 2. Does there exist a graph Y and c > O such that x(Y)=w,, lYl=ol and f A n ) >cn. Here we only have the very little information that the Specker graph Y l ( o l , 3) does not have this property. To see this it is sufficient to prove:

Tbeorem 2. Let Y = Y,(w, 3). Then

Proof. We define an B c . [ n l 3 with lqa n log n/8 log log n such that for all M c 9, M independent in 3 ' , IM(<4n holds for all sufficiently large n. This will prove the result. Let ai = [log nilfor i < io = [log n/log log n ] . Then ai E n for i < i o . Let X ( t , i ) = ( t , ? + a z i ,t + a 2 i + l ) and B={X(t, i ) : X(t, i)cn}. Then ( B ( > , nlog n/8 log log n provided n is large enough. Let M c $ JMI34n. We prove that M contains an edge of

Almost bipartite large chromatic graphs

Y.Let Mi= { t : X(t, i ) E M ) . Then

121

JMil= IMI 24n. Let

N i = { t € M i : 3 t ' ~ M r'+azi
We claim that [Nil2 [Mil- 2n/log n. We can choose elements to, . . . ,tj- of Mi so that M i c U { [ t l , t r + a 2 i + l ) :l < j ) where the above intervals are pairwise disjoint. Now M i \ N i c IJ{ [trt tr+azi+ I]\(tl+a2i, tl+a2i+ 1): l. Since a2i 2 -<-, a ~ i + l logn

/Mi- Nil <2n/log n. It follows now, that

c INiI 2 C IMiI

i < io

i < io

-

2n log log 3 4 n -

2n >n. log log n

-~

Then there are i l < i2 with NilnNi,#O. Let t E NilnN i , . Then for some t' E Mi,, t ' + ~ ~ ~ , < t < t ' + a ~X(t', ~ , +il) ~ ,E M. On the other hand X(t, i 2 ) E M i , c M . Now t + a ~ i 2 > t ' + a ~ i 2 > t ' + ~ 2since i l + i [log n2i1+1]<[logn 2 ' 2 ] if i l < i 2 and n is large enough. Then X(t', i l ) and X(t, i2) are adjacent in Y.

3. Omitting edges of a subgraph

Delinition 3.1. Let ,f&n)=max{min{lE'1: ( A , [A]'nY\E')

is bipartite): ACT/ ~ l A ( = n } ;

more generally for 2
k') for k' < k .

It is also clear from the definition that n -f;(n)< 2f&1). This in view of Lemma 2.1 implies that for any given graph Y with x(Y) > w there is an E > 0 such thatfi(n) 3 En. This contrasts again with the situation for finite graphs. L. Lovasz recently informed us that, as a generalization of a theorem of T. Gallai, he can prove the following theorem: For 2 < r < o there is a j n i t e graph Y with x(Y)>r+2 andf&)=O(n'-'"). We describe his example: Let rn be even. Let V be the set of r-dimensional lattice points mod(rn),(xo, . . . ,x, - (yo, . . . ,y, - ) are connected if for some i < r [ x i- yil = 1, and x j = y j for j # i , j < r . This graph satisfies the requirements for large enough rn. In case of infinite chromatic graphs we are again left with the examples Yo(cc, k).

P . Erdos e l al.

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Theorem 3. (a) For all K > W there exists a graph with

~ ( Y ) > K and f & 1 ) < 2 n ~ ’ ~ (b)V E > Othere i s an r < O such that for all K > O there exists a graph with x ( S ) >K undf;(n, r)
By Lemma 1.1 it is sufficient to prove

Theorem 3.A. (a) If Y= Yo(w, 2), t h e n f & 1 ) < 2 n ~ ’ ~ . (b) I f Q= Y(w k ) for some 3 < k 0 there is an r < O such [hat f:(O,

r)
Proof. (a) Let V c [ o l 2 , IVI=n. Put V ( x ) = { { x y, ) : {x, y ) E V ) , v ( x ) = ( V ( x ) I forx E O . Fore E Klet Q(e, V‘)= {e’E V’:{e,e’}E 48). Let A = { x E O : u ( x ) 2 n ” 2 } .Then (Al
Iu

Theorem 4. Let 2 < k < o . Y is said to have property P ( k ) iffor all q > O there is an r < w w i t h j i ( n , r ) < n l + k - l + oIf. Y=(K E ) has property P ( k ) and < is any ordering of V, then OE(Y, <)has the property P(k + 1).

Proof. Let r(q)=min(r: f&,ri
f&&,

s) < n

l + ( k + l -1+q.

Given q>O, first choose q k q , q ’ > O . Then choose q”>O so that (1 - ( k + l)-’-q’). ( 1 + k - ’ + q “ ) = 1-6 for some 6>0. Let 1 be an integer with 1
E

V let E’(x)=

Y(e, E”)= {e‘ E E”: {e, e’) E OE(3, <)}. Let A = { x E V : e ’ ( x ) a n ( k + l ) - l f ” Then ‘ ) . l A l < n l - ( k + l ) - l - q ’ . We partition the set E‘ into three pieces. Let E j = { e E E‘: Ie-AI=j} for j < 3 . For each e E E‘, I3(e, E2)l< 2n(k+1)-1+q’, hence omitting 2n’+(k+1)-1+q’ edges. E 2 is independent and no edge

Almost bipartite large chromatic graphs

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joins any point of E 2 to any other point of E'. The subgraph spanned by El in the ordered edge graph is clearly 2-chromatic, Eo G[A]'. Since (A1
Problem 3. Assume Y has chromatic number w . Can f:(n) slowly? Can it be at most log n or log(k)(n)for k < w ?

tend t o infinity very

Referem [l] P. Erdos, Graph theory and probability, Canad. J. Math. I I (1959) 34-38. [2] P. Erdos, O n circuits and subgraphs of chromatic graphs, Mathematica 9 (1962) 17C175, [3] P. Erdos, F. Galvin and A. Hajnal, On set-systems having large chromatic number and not containing

prescribed subsystems, in: Colloquia Math. SOC. Janos Bolyai l c l n f i n i t e and finite sets (Keszthely, Hungary, 1973) pp. 425-513. [4] P. Erdos and A. Hajnal, On chromatic number of graphs and set-systems, Acta Math. Acad. Sci. Hung. 17 (1966)61-69. [5] P. Erdos, A. Hajnal and S. Shelah, On some general properties of chromatic numbers, in: Colloquia Math. Soc.Janos Bolyai 8-Topics in Topology (Keszthely, Hungary, 1972) pp. 243-255. Received 19 December 1979.