Some Geometric Applications of Dilworth s Theorem

Transcription

1 Some Geometric Applications of Dilworth s Theorem J~NOS PACH* JENO TOROCSIK Hungarian Academy of Sciences and Mathematical Institute Dept. Comp. Sci., City College, New York of the Hungarian Academy of Sciences pach@cims6.nyu.edu ecomool@ursus.bke.hu Abstract on partially ordered sets. A geometric graph is a graph drawn in the plane such that its edges are closed line segments and no 3 vertices are collinear. We settle an old question of Avital, Hanani, Erdos, Kupitz and Perles by showing that every geometric graph with n vertices and m > k4n edges cent ains k+ 1 pairwise disjoint edges. We also prove that, given a set of points V and a set of axis-parallel rectangles in the plane, then either there are k + 1 rectangles such that no point of V belongs to more than one of them, or we can find an at most ks element subset of V meeting all rectangles. This improves a result of Ding, Seymour and Winkler. Both proofs are based on Dilworth s theorem *Research supported by Hungarian National Foundation for Scientific Research Grant OTKA-1412 and NSF Grant CCR Introduction Ever since Erdos and Szekeres [ES] rediscovered Ramsey s theorem [R] to show that out of ~:24 points in the plane it is possible to select () n points which form a convex n-gon, combinatorial geometry has been a rich area of application for Ramsey theory (see [GRS] ). Many classes of geometric objects can be equipped with natural ordering relations, and this information is often lost when we apply Ramsey s theorem. For example, given n convex bodies in the plane, Ramsey)s theorem only implies that we can choose ~ log2 n of them which are either pairwise disjoint or pairwise intersecting. However, as we have shown in a recent paper [LMPT], one can also choose nl/5 elements with the above property. Our proof was based on the following wellknown result [D]. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM corwright notice and the title of the publication and ite data appsar, and notice is given that copying is by permission of the Aeeociation for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 9th Annual Computational Geometry,5/93/CA, USA a 1993 ACM /93 /0005 / $1.50 Dilworth s Theorem Let P be a parizalzy ordered set containing no chain (totally ordered subset) of k + 1 elements. Then 1 can be COVered by k antichains (subsets of pairwise incomparable elements). 264

2 In this note we shall present some further applications of the same idea. A geomet~ic graph G is a graph drawn in the plane by (possibly crossing) line segments, i.e., it is clefinecl as a pair (V(G), E(G)), where V(G) is a set of points in the plane in general position and E(G) is a collection of closed line segments whose endpoints belong to V(G). The following question was raised by Avital ancl Hanani [AH], Kupitz [K] ancl Perles: Determine the smallest number e~(n) such that any geometric graph with n vertices and m > e~(n) edges cent ains k + 1 pairwise disjclint edges. An old result of Hopf and Pannwitz [HP] and Erdos [E] implies that e,(n) = n. Alon and Erclos [AE] showed that e2(n) ~ 6n, which was subsequently improvecl by O Donnel ancl Perles [OP] and Goddard, Katchalski and Kleitman [GKK] to ez(n) < 3n. In the latter paper it has also been established that e3(7z) < 10n ancl, for any fixed k, e~(n) < ckn log~-3 n with a suitable constant c~. Theorem 1 Any geometric graph with n vertices and more than k% edges contains at least k + 1 pairwise disjoint edges. This means that ek(n) s k4n for all k and n. Moreover, we can show that if a geometric graph has much more than e.~(n) edges then it has many (k + 1)-tuples of pairwise clisjclint eclges. Theorem 2 J o~ eve~y k the~e ezists CL :> 0 such that any geometrzc graph with n vertices and m > k4n edges has at least c~m2~f~/nz~ (k+ 1)-tuples of pairwise disjoint edges. PuTihermore, this bound is asymptotically tight apart from the exact value of c~. Our next result can also be established by repeated application of Dilworth s theorem. It is a substantial improvement of a theorem of Ding, Seymour and Winkler [DSW], whose original proc)f requires moire involved techniques. Theorem 3 Let V and R be a set of poznt.~ and a set of axis-parallel rectan~gles in the plane, respectively. Then for any natural number k, either (i) (ii) there are k + 1 rectangles in 72 so that no point of V belongs to more than one of them, or one can choose at most k8 points of V so that every member of 7? contains at least one of them. In [DSW] the same result was proved with (k+ 63)127, instead of k8. 2 Geometric Graph/s First we prove Theorem 1. Let G be a geometric graph on n vertices and m edges, containing no k + 1 pairwise disjoint edges. Let x(v) and y(v) denote the ~-coordinate and the y- coordinate of a point v, respective] y. For any two edges e = 7J1vZ, e = vjvj G E(G), we say that e precedes e (in notation, e << e ) if X(vl) ~ Z(V( ) and X(V2) < z(vj). Furthermore, e is said to lie below e, if there is no vertical line 1 (parallel to the y-axis) which intersects both e and e with Y(1 n e) 2 y(l n e ). If e is below e, than we shall write e < e. Note that < is not necessarily a transitive relation on E(G). Finally, let ~(e) denote the orthogonal projection of e onto the z-axis. 265

3 Define four binary relations <i (i = 1,...4) on E(G), as follows. Two eclges can be relat ecl by any of these relations only if they are disjoint. Given two disjoint edges e, e c E(G), let e <1 et if e << e and e < e, e~ze if e << e ancl e > e, e <3 e if 7r(e) ~ 7r(e ) ancl e < e, c <4 e if m(e) S 7r(e ) ancl e > e. It follows reaclily from the clefinitions that (a) each of the relations <; (i = 1,...4) is transitive, (b) any pair of disjoint edges is compared by at least one of the relations -+ (i = 1,..., 4). None of the partial orders (13(C), <;) contains a chain of length k+ 1, otherwise G would have k + 1 pairwise disjoint edges. By Dilworth s theorem, for any Z, E(G) can be parti tioned into at most k classes so that no two eclges belonging to the same class are compared by <i. Superimposing these four partitions, we obtain a decomposition of E(G) into at most k4 classes Ej (1 < j < k4) so that no two elements of Ej are related by any <~. Hence, none of the classes Ej contains two disjoint edges, which implies that ]Ej/ < cl(n) = n. Therefore, m = IE(G)I Il?j I s k n, as desired. l edges, let ~(G) denote the number of k + 1- tuples of pairwise disjoint edges in G. We are going to show, by induction on n, that (*) f(g) > (6k)- whenever m ~ e~(n) + n. (2;+2)(5)2 + It follows from Theorem 1 that f(g) > m ek(n), which is stronger than (*) if e~(n) + n s m s e~(n) + 2n, and it also shows that (*) holds for n = k4 +3. Suppose now that n > k4+3, m > e~(n)+2n and that we have already established the above inequality for all graphs with at most n 1 vertices. For any v c V(G), let d(v) clenote the number of edges of G incident to v, and let G v stand for the graph obtained from G by the deletion of v. Using the induction hypothesis, we get (n - 2k - 2)f(G) = ~ f(g -v) ~ (6k)- k () n 1 ~::+:k+l () 2 v6v(g) ~ (m - d(.)) + VCV(G) (Notice that G v has at least m (n 1) 2 e~(n 1) + (n 1) edges.) However, by.jensen s inequality, 2( m cl(v))2k+1 > v~v(g) Next we deduce Theorem 2 from Theorem 1, by using an idea of Ajtai, Chv6.tal, Newborn ancl Szenler6di [ACNS]. Given a geometric graph G with n vertices and m 2 e~(n) + n ~b 2k 2k+1 (n_ ~)2k+lm2k+l x (~ - d(v)) = ~2k? ( VEV(G) ) 266

4 and (*) follows. This proves the first part of Theorem 2. To show that for any fixed k the bound is tight up to a constant factor clepending OCJy on k, assume that n s m ~ n2/16 and set ~ = \&J. Divide a circle into 2t equal arcs Al,...A2t, and pick n vertices on the circle as equally clistributed among the arcs as possible..join every vertex belonging to Ai to all vertices in Ai+~ by line segments (1 < z < t). The resulting geometric graph has 72 vertices and at least m eclges. Furthermore, for any (k + I.)- tuple of pairwise clisjoint eclges of G, there exists an i such that all endpoints of these edges lie in Ai U A;+t. Thus, the number of these (k+ 1)-tuples cannot exceed on %2, as follows. Two rectangles can be related by any of these relations only if they are almost disjoint. Given a pair of almc)st disjoint rectangles R, S C R., let R<3S if It follows from the definitions that 72 z~+z < (lotn) t [jjl ~+ /r12k, completing the proof of Theorem 2. R 3 Systems of points and rectangles In this section we are going to establish Theorem 3. Let V be a set of points and R a set of axisparallel rectangles in the plane. Two rectangles R, S c 7? are said to be almost disjoint if R n S contains no element of V. For any R c XI, let 7rz(R) and TV(R) denote the orthogonal projections of R onto the z-axis and onto the y-axis, respectively. Given two intervals 1, J on the x-axis (or on the y-axis), we say that 1 precedes J (or, in notation, I << J) if the left and right endpoints of 1 precede the left ancl right endpoints of J, respectively. Define eight binary relations <i (i = 1,...8) (a) each of the relations <i (i = 1,...8) is transitive, (b) any pair of almost clisjoint rectangles is comparable by one of the relaticms <i (i = 1 ~...? 8). We can assume without loss of generality that none of the partial orders (R, <~) cent ains a chain of length k +1, otherwise condition (i) of the theorem holds. Hence, by thle repeated application of Dilworth s theorem, we obtain that X can be partitioned into k8 classes l+ (1 < j s k ) so that no pair of rectangles bel;nging to the same class is comparable by any of the relations <i. This means, by (b), that the intersection of any two rectangles belonging to the same class contains at least one point of V. To prove the theorem, it is sufficient to show that for any j one can select at most points of V such that every member of %3j contains at least one of them. 267

5 Lemma 1 Let V be a set of points, and let 7?0 be a set of axis-parallel rectangles in the plane such that the intersection of any two members contains at least one element of V. Then one can choose at most points of V so that every member of 7?0 contains at least one of them. ProoE By Helly s theorem, (-lro # 0. Thus, we can assume without loss of generality that the origin (O, O) is contained in ever y member of %?O. Suppose, for simplicity, that no point of the coordinate axes belongs to V. Let ~ denote the set of all points of V lying in the i-th quadrant of the coordinate system (i = 1,... 4). Let ~~ consist of all points (z, y) e Vi for which there is no (x, y ) 6 U with x c [0, z], y c [0, y]. Finally, set V = V{ U... U Vi. Notice that (a) if we number the elements of W according to their (say) z-coordinates, then the intersection of any rectangle R 6 T& with ~ consists of a single interval of consecw tive points (i = 1,....4); (b the intersection of any two members of RO contains at least one element of V. l_ general theorem of Gy&f& and Lehel [GYL] states that, for every p and q, there exists ~(p, q) < co with the following property. Let.F be a family of sets, each of them obtainecl as the union of p intervals taken from p parallel lines. If.F does not have more than q pairwise disjoint members, then one can find at most f(p, q) points so that every member of F contains at least one of them. This easily implies, by (a) ancl (b), that V has a subset of size ~(4, 1) ~ meeting every member of %&. 4 Concluding remarks Given a partially ordered set of m elements, in 0(rn2) time one can perform a topological sort and find a maximal chain (cf. [CLR] ). Hence, the proof in Section 2 also yields the following result. Corollary 1 There is an 0(m2)-izme,algorithm for finding at least (m/n) l/4 pairwise disjoint edges in any geometric graph with n vertices and m edges. Note that the proof of Theorem 3 can be easily extended to higher dimensions. Theorem 4 Let V and B be a set of points and a set of axis-parallel boxes in Rd, respectively. Then, for any natutal number k, either (i) there are k + 1 boxes in B so that no point of V belongs to more than one of them, or (ii) one can choose at most cdk22d- points in V so that any member of B contains at least one of them, where cd is a constant depending only on d. Using some related techniques we can also establish the following strengthening of a theorem of B&rAny and Lehel. Given two points ~, y e Rd, let 130x(z, y) denote the smallest box parallel to the axes, which contains z and Y. Theorem 5 Any set of points V & Rd contains at most 22 elements Xi (1 < i < 22 ) such that IJ Box(~i, Zj) 2 v. l<i<j~22d 268

6 Acknowledgment We thank.j. Matou5ek for his helpful suggestions, References - \cikkj W. Goddard, M. Katchalski and D. J,. Kleitman: Forcing disjoint segments in the p/ane, manuscript. [GRS] R. L. Graham, B. L. Rothschild ancl J. H. Spencer: Ramsey Theory (2nd edition), J. Wiley and Sons, New York, [ACNS] M. Ajtai, V. Chv6tal, M. Newborn and E. Szemer4di: Crossingfree.mbgraph.s, Ann, Discrete Math. 12 (1982), [GYL] A. Gy&rf&s, J. Lehel: Covering and coloring problems for relatives of intervals, Discrete Math. 55 (1985), [AE] N. Alon, P. Erdos: Disjoint edges in geometric graphs, Discrete and Comput. Geom. 4 (1989), [HP] H. Hopf and E. Pannwitz: Aufgabe No. 167, Jber. Deutch. Math Verein, 43 (1934), 114. [AH] [CLR] S. Avital and H. Hanani: Graphs, Gzlyonoi Lematematika 3 (1966), 2-8 (in Hebrew). Th. H. Cormen, Ch. E. Leiserson and R. L. Rivest: Introduction to Algorithms, MIT Press, Cambridge and McGraw-Hill, New York, [K] Y. S. Kupitz: Extremal Problems in Combinatorial Geometry, Aarhus Universit y Lecture Notes Series, No. 53, Aarhus University, Denmark, [LMPT] D. Larman, J. Matou3ek, J. Path and J. Tor8csik: A Ramsey-type Tesult for planar convex sets, to appear. [D] R. P. Dilworth: A decomposition theorem for partially ordered sets, Ann. [OP] P, O Donnel and M. Perles: Every geometric graph with n vertices and Math. 51 (1950), n i- 3.4 edges contains 3 pairwise [DSW] G. Ding, P. Seymour and P. Winkler:.Bounding the vertex covet numbet of a hypergraph, to appear in Combinatorics. [R] disjoint edges, manuscript, Rutgers University, New Brunswick, F. P. Ramsey: On a probzem of formal logic, Proc. London Math. Sot. 30 [E] P. Erdos: On sets of distances of (1930), n points, Amer. Math. Monthly, 53 (1946), [ES] P. Erdos and G. Szekeres: A combinatorial ptoblem in geometry, Composition Math. 2 (1935). \ /,