Ramsey-Type Results for Unions

of Comparability Graphs

Adrian Dumitrescu1� and Geza Toth2y

1 Computer Science, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA. e-mail: dumitres@paul.rutgers.edu2 Department of Mathematics, Massachusetts of Technology, 77 Massachusetts Avenue,Cambridge, MA 02139-4307, USA and Hungarian Academy of Sciences, Budapest,Hungary. e-mail: toth@math.mit.edu

Abstract. It is well known that the comparability graph of any partially ordered set of nelements contains either a clique or an independent set of size at least

ffiffiffin

p. In this notewe show

that any graph of n vertices which is the union of two comparability graphs on the same vertexset, contains either a clique or an independent set of size at least n

13. On the other hand, there

exist such graphs for which the size of any clique or independent set is at most n0:4118. Similarresults are obtained for graphs which are unions of a fixed number k comparability graphs.We also show that the same bounds hold for unions of perfect graphs.

1. Introduction

Let S ¼ ðV ;�Þ be a partially ordered set on V ¼ fv1; v2; . . . ; vng. The compara-bility graph of S, G ¼ ðV ;EÞ is a graph such that ðvi; vjÞ 2 E if and only if eithervi � vj or vj � vi.

A graph G is called perfect, if for any spanned subgraph G0 � G, its chromaticnumber is equal to the size of its largest clique (see also [2], [9]).

Dilworth’s Theorem. [4] Let S be a partially ordered set containing no chain (totallyordered subset) of size k þ 1. Then P can be covered by k antichains (subsets ofpairwise incomparable elements).

It follows directly from Dilworth’s theorem that comparability graphs areperfect. Any perfect graph of n vertices contains either a clique or an independent

� Supported by DIMACS Center, Rutgers Universityy Supported by NSF grant DMS-99-70071, OTKA-T-020914 and OTKA-F-22234. Presentaddress: A. Renyi Institute of Mathematics, POB 127, H-1364 Budapest, Hungary. e-mail:geza@renyi.hu

Graphs and Combinatorics (2002) 18:245–251

Graphs andCombinatorics� Springer-Verlag 2002

set of size at leastffiffiffin

p. It is easy to see that this bound can not be improved even

for comparability graphs.In this note we obtain similar results for unions of two or more comparability

(resp. perfect) graphs on the same vertex set.

Definition. Let fkðnÞ (resp. gkðnÞ) be the largest number such that any graph G of nvertices, which is the union of k comparability graphs (resp. perfect graphs) on thesame vertex set, contains either a clique or an independent set of size at least fkðnÞ(resp. gkðnÞ).

Theorem 1.

n1

kþ1 � gkðnÞ � fkðnÞ � n1þlog k

k :

In particular, for k ¼ 2,

Theorem 2.

n13 � g2ðnÞ � f2ðnÞ � n0:4118:

Unions of comparability graphs are often used in combinatorial geometry, forproving Ramsey-type results ([10], [13], [8], [7], [17], [14], [16], see also theRemarks).

Let xðGÞ, aðGÞ, and vðGÞ denote the clique number, independence number,and the chromatic number of a graph G, respectively. Obviously, n=aðGÞ � vðGÞ,therefore the inequality n

1kþ1 � fkðnÞ is a direct consequence of the following

stronger result, which is essentially tight.

Theorem 3. If G is the union of k perfect graphs, then

vðGÞ � xkðGÞ:

On the other hand, for any k there is a graph G which is the union of k comparabilitygraphs, with

vðGÞ � xðGÞð Þkð1�oð1ÞÞ:

2. Proof of Theorems 1 and 2

All comparability graphs are perfect [9], therefore gkðnÞ � fkðnÞ.We prove the lower bound by induction on k. For k ¼ 1, the statement is a

direct consequence of the definition of perfect graphs [9]. Suppose that we havealready proved the statement for k � 1 and for all n. Let GiðV ;EiÞ (1 � i � k) be

246 A. Dumitrescu and G. Toth

perfect graphs, V ¼ fv1; v2; . . . ; vng. Suppose for simplicity that m ¼ n1=ðkþ1Þ is aninteger. Suppose that the size of any clique in GðV ;EÞ, E ¼ [n

i¼1Ei, is less than m.Since G1ðV ;E1Þ is perfect and xðG1Þ < m, it can be colored by less than mcolors, so it contains an independent set of size at least n=m ¼ mk, say,V 0 ¼ fv1; v2; . . . ; vmkg.

For i ¼ 2; . . . ; k, let G0i be the subgraph of Gi spanned by V 0. The graphs Gi are

perfect, hence G0i are also perfect. By the induction hypothesis and the assump-

tion, there is a set of size m, which is an independent set in each G0i, i ¼ 2; . . . ; n.

But then it is an independent set in GðV ;EÞ, proving the lower bound.

Definition. Let G1ðV1;E1Þ and G2ðV2;E2Þ be two (directed ) graphs. The orderedproduct GðV ;EÞ ¼ G1 � G2 is a (directed ) graph with vertex set V ¼ V1 � V2, andedge set

EðGÞ ¼ ðx1; x2Þ; ðy1; y2Þð Þ j x1; y1 2 V1; x2; y2 2 V2; andfeither ðx1; y1Þ 2 E1 or x1 ¼ y1 and ðx2; y2Þ 2 E2g:

Let k be a fixed number. By known bounds for Ramsey numbers [1], we knowthat there is a graph G such that jV ðGÞj ¼ 2k, and xðGÞ; aðGÞ < 2k.

Proposition. Every graph G is the union of dlog vðGÞe bipartite graphs.

Proof. Take a vðGÞ-coloring of G. Clearly, there exists a bipartite graph B1 � Gon V , such that the chromatic number of GnB1 is at most dvðGÞ=2e. Similarly, wecan take a bipartite graph B2 � GnB1, such that the chromatic number ofGnB1nB2 is at most dvðGÞ=4e. After dlog vðGÞe analogous staps, the remaininggraph is 1-colorable, that is, we decomposed the edge set of G into dlog vðGÞebipartite graphs.

By the Proposition, the edge set of G can be decomposed into k bipartitegraphs. Since bipartite graphs are comparability graphs, G is the union of kcomparability graphs. Let

Gi ¼ G� � � � � Gzfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{i times

:

It is easy to see that Gi is also a union of k comparability graphs. jV ðGiÞj ¼ 2ik ,and

xðGiÞ; aðGiÞ < ð2kÞi ¼ jV ðGiÞj1þlog k

k :

Now we show the upper bound of Theorem 2. We define two graphs, H1 andH2. V ðH1Þ ¼ fv1; v2; . . . v13g, ðvi; vjÞ 2 EðH1Þ if and only if

i� j � 1; 5; 8; or 12 ðmod 13Þ:

Ramsey-Type Results for Unions of Comparability Graphs 247

See Figure 1. By Brooks’ theorem [3] H1 is four-colorable, therefore by theProposition, it is the union of two comparability graphs. Such a decomposition isgiven on Figure 1.

It is easy to see that xðH1Þ ¼ 2 and aðH1Þ ¼ 4.For H2, consider the set S of 12 chords of a cycle, shown on Figure 2.

There are no three pairwise intersecting and no five pairwise disjoint chords inS. The vertices of H2 represent the chords in S, and two vertices are con-nected if and only if the corresponding chords are disjoint. Then jV ðH2Þj ¼ 12,xðH2Þ ¼ 4 and aðH2Þ ¼ 2. We show that H2 is the union of two comparabilitygraphs. Suppose the endpoints of the chords in S are denoted by 1; 2; . . . ; 24,in clockwise direction (starting with an arbitrary endpoint), ða; bÞ; ðc; dÞ 2 S,a < b, c < d. Then let ða; bÞ �1 ðc; dÞ if and only if a < c < d < b, and letða; bÞ �2 ðc; dÞ if and only if a < b < c < d. Clearly, both �1 and �2 are partialorderings, and two chords are comparable by one of them if and only ifthey are disjoint. This implies a decomposition of H2 into two comparabilitygraphs.

Let G ¼ H1 � H2. G has 156 vertices, xðGÞ ¼ aðGÞ ¼ 8, G is the union of twocomparability graphs. Finally, let

Gi ¼ G� � � � � Gzfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{i times

:

Then Gi is still the union of two comparability graphs, jV ðGiÞj ¼ 156i, and

xðGiÞ; aðGiÞ ¼ 8i ¼ jV ðGiÞjlog 8log 156 < jV ðGiÞj0:4118:

Fig. 1

248 A. Dumitrescu and G. Toth

3. Proof of Theorem 3

Suppose thatG is the union of k perfect graphs,G1;G2; . . . ;Gk, all on the same vertexset, and letm ¼ xðGÞ. Since eachGi can be colored bym colors, the product of thesecolorings is a proper coloring of G, with at most mk ¼ xkðGÞ colors.

To establish the second statement of the theorem, take a triangle-free graph Hwith t vertices and aðHÞ �

ffiffit

plog t. The existence of such a graph was proved by

Erdo}s [5].

Claim. For a triangle-free graph G, vðGÞ � aðGÞ þ 1.

Proof of Claim. Let DðGÞ be the maximum degree in G. For any graph,vðGÞ � DðGÞ þ 1, and since G is triangle-free, the neighborhood of any vertex isan independent set, consequently, DðGÞ þ 1 � aðGÞ þ 1.

By the Claim, vðHÞ �ffiffit

plog t þ 1. Therefore, by the Proposition, we can de-

compose H into logðvðHÞÞd e � ð1=2þ oð1ÞÞ log t bipartite graphs. Since all bi-partite graphs are comparability graphs, H is the union of k ¼ ð1=2þ oð1ÞÞ log tcomparability graphs. Let

Gr ¼ H � � � � � Hzfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{r times

:

Gr has tr vertices, xðGrÞ ¼ 2r, aðGrÞ � ðffiffit

plog tÞr and vðGrÞ �tr=aðGrÞ �

ðffiffit

p= log tÞr. Since the product of comparability graphs is also a comparability

graph, Gr is the union of k ¼ ð1=2þ oð1ÞÞ log t comparability graphs. Thus, wehave,

vðGrÞ � ðffiffit

p= log tÞr � xðGrÞð Þkð1�oð1ÞÞ:

Fig. 2

Ramsey-Type Results for Unions of Comparability Graphs 249

4. Remarks

Very recently Theorem 2 has been improved by Tibor Szabo, showing thatf2ðnÞ � n0:3878 [15].

Probably the first Ramsey-type result in geometry that used unions of com-parability graphs is the result of Larman et al.

Theorem [10]. Among any n convex sets in the plane, there are either n1=5 pairwiseintersecting or n1=5 pairwise disjoint.

This result is very likely not the best possible. The best known result from theother direction is given by Karolyi et al.

Theorem [8]. For any n large enough, there exists a collection of n convex sets in theplane such that no n0:4207 of them are pairwise intersecting or pairwise disjoint.

For some special classes of convex sets there are even stronger results.

Theorem [6], [10]. Any collection of n axis-parallel rectangles containsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin= log n

p

pairwise intersecting or pairwise disjoint.

Theorem [7]. Any collection of n chords of a circle contains 1þ oð1Þð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2n= log n

p

pairwise intersecting or pairwise disjoint.

Theorem [14]. Among any n translates of a convex set in the plane there are eitherc

ffiffiffin

ppairwise intersecting or c

ffiffiffin

ppairwise disjoint.

It is the consequence of the following more general result. We say that aconvex planar set K-fat, if the ratio of the radii of the smallest covering disc andthe largest inscribed circle R=r < K.

Theorem [11]. For any K > 0, any collection of n K-fat convex sets in the planecontains cK

ffiffiffin

ppairwise intersecting or pairwise disjoint.

A geometric graph is a graph drawn in the plane so that the vertices arerepresented by points in general position, the edges are represented by straight linesegments connecting the corresponding points. Using comparability graphs, Pachand Toro}csik obtained the following result.

Theorem [13]. Any geometric graph of n vertices and more than k4n edges containsk þ 1 pairwise disjoint edges.

Using similar methods, this bound was improved in [17] and recently furtherimproved in [16].

Theorem [16]. Any geometric graph of n vertices and more than 29k2n edges con-tains k þ 1 pairwise disjoint edges.

250 A. Dumitrescu and G. Toth

Acknowledgments. We are very grateful to Janos Pach and Michael Saks for their com-ments.

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Received: November 1, 1999Final version received: December 1, 2000

Ramsey-Type Results for Unions of Comparability Graphs 251