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Coloring K K-Free Intersection Graphs of Geometric Objects in the Plane

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Coloring K K-Free Intersection Graphs of Geometric Objects in the Plane Coloring Kk-free intersection graphs of geometric objects in the plane ¤ y Jacob Fox János Pach Department of Mathematics Dept. Computer Science Princeton University City College of New York Princeton, NJ 08544, USA New York, NY 10031, USA [email protected] [email protected] ABSTRACT size of the largest clique, and the chromatic number Â(G) The intersection graph of a collection C of sets is a graph of G is the minimum number of colors needed to properly color the vertices of G. To compute or to approximate these on the vertex set C, in which C1;C2 2 C are joined by an parameters is a notoriously di±cult problem [18, 35, 23]. In edge if and only if C1 \ C2 6= ;. Erd}os conjectured that the chromatic number of triangle-free intersection graphs of n this paper, we study some geometric versions of the question. segments in the plane is bounded from above by a constant. The intersection graph G(C) of a family C of sets has ver- Here we show that it is bounded by a polylogarithmic func- tex set C and two sets in C are adjacent if they have non- tion of n, which is the ¯rst nontrivial bound for this problem. empty intersection. The independence number of an inter- More generally, we prove that for any t and k, the chromatic section graph G(C) is often referred to in the literature as the packing number of C. It is well known that the problem number of every Kk-free intersection graph of n curves in the plane, every pair of which have at most t points in common, of computing this parameter, even for intersection graphs of log n c log k families of very simple geometric objects such as unit disks is at most (ct ) , where c is an absolute constant log k or axis-aligned unit squares, is NP-hard [17, 25]. Because of and ct only depends on t. We establish analogous results for intersection graphs of convex sets, x-monotone curves, its applications in VLSI design [24], data mining [9, 26], map semialgebraic sets of constant description complexity, and labeling [3], and elsewhere, these questions have generated sets that can be obtained as the union of a bounded number a lot of research. In particular, starting with the work of of sets homeomorphic to a disk. Hochbaum and Maas [24], several polynomial time approxi- Using a mix of results on partially ordered sets and planar mation schemes (PTAS) have been found in special settings separators, for large k we improve the best known upper [3, 9, 10]. bound on the number of edges of a k-quasi-planar topological Motivated by applications in graph drawing and in geo- graph with n vertices, that is, a graph drawn in the plane metric graph theory, here we establish lower bounds for with curvilinear edges, no k of which are pairwise crossing. the independence numbers of intersection graphs of fami- As another application, we show that for every " > 0 and lies of curves in the plane. Following [41], some algorithmic for every positive integer t, there exist ± > 0 and a positive aspects of this approach were explored in [4]. Obviously, ®(G) ¸ n=Â(G) holds for every graph G with n vertices. integer n0 such that every topological graph with n ¸ n0 vertices, at least n1+" edges, and no pair of edges intersecting Therefore, any upper bound on the chromatic number yields in more than t points, has at least n± pairwise intersecting a lower bound for the independence number. It will be more edges. convenient to formulate our results in this more general set- ting. The study of the chromatic number of intersection graphs 1. INTRODUCTION of segments and their relatives in the plane was initiated by For a graph G, the independence number ®(G) is the size of Asplund and GrunbaumÄ [7] almost half a century ago. Since the largest independent set, the clique number !(G) is the then, this topic has received considerable attention [5, 22, ¤ 27, 30, 31, 32, 36, 44]. In particular, a classical question Supported by an NSF Graduate Research Fellowship and of Erd}os [21, 31, 36] asks whether the chromatic number of a Princeton Centennial Fellowship. y all triangle-free intersection graphs of segments in the plane Supported by NSF Grant CCF-05-14079, and by grants is bounded by a constant. It is know that there exist such from NSA, PSC-CUNY, the Hungarian Research Founda- graphs with chromatic number eight. In the ¯rst half of this tion OTKA, and BSF. paper, we provide upper bounds on the chromatic number of intersection graphs of families of curves in the plane in terms of their clique number. In particular, we prove that Permission to make digital or hard copies of all or part of this work for every triangle-free intersection graph of n segments in the personal or classroom use is granted without fee provided that copies are plane has chromatic number at most polylogarithmic in n. not made or distributed for profit or commercial advantage and that copies Most of our results generalize to intersection graphs of fam- bear this notice and the full citation on the first page. To copy otherwise, to ilies of planar regions whose boundaries do not cross in too republish, to post on servers or to redistribute to lists, requires prior specific many points (e.g., semialgebraic sets of bounded description permission and/or a fee. complexity) and to families of convex bodies in the plane. SCG’08, June 9–11, 2008, College Park, Maryland, USA. Copyright 2008 ACM 978-1-60558-071-5/08/04 ...$5.00. See Subsection 1.1. In the second half of the paper, we apply our results Theorem 3. If G is a Kk-free intersection graph of a t- to improve on the best known upper bounds on the max- intersecting family of n r-regions, then imum number of edges of k-quasi-planar topological graphs. cr log k The terminology and the necessary preliminaries will be ex- log n Â(G) · ct;r ; plained in Subsection 1.2. log k 1.1 Upper bounds on the chromatic number where ct;r only depends on t and r and c is an absolute of intersection graphs constant. A (simple) curve in the plane is the range of a continuous A semialgebraic set in Rd is the locus of points that satisfy (bijective) function f : I ! R2 whose domain is a closed in- a given ¯nite boolean combination of polynomial equations terval I ½ R. A family of curves in the plane is t-intersecting and inequalities in the d coordinates. The description com- if every pair of curves in the family intersect in at most t plexity of such a set S is the minimum · such that there is a points. representation of S with dimension d at most ·, the number The following theorem gives an upper bound on the chro- of equations and inequalities at most ·, and each of them matic number of the intersection graph of any t-intersecting has degree at most ·. (See [8].) family of n curves with no clique of order k. As mentioned in [15], every semialgebraic set in the plane of constant description complexity is the intersection graph Theorem 1. If G is a K -free intersection graph of a t- k of a t-intersecting family of r-regions, where r and t depend intersecting family of n curves in the plane, then only on the description complexity. Therefore, we have the log n c log k following corollary of Theorem 3. Â(G) · c ; t log k Corollary 4. If G is a Kk-free intersection graph of a 2 where ct is a constant in t and c is an absolute constant. family of n ¸ k semialgebraic sets in the plane of descrip- tion complexity d, then In other words, for every family C of n curves in the plane with no pair intersecting in more than t points and no k log n cd log k Â(G) · ; curves pairwise crossing, each curve can be assigned one of log k c log k log n at most ct colors such that no pair of curves of log k where cd is a constant that only depends on d. the same color intersect. Here, and throughout the paper, unless it is indicated otherwise, all logarithms are assumed An x-monotone curve is a curve in the plane such that to be to the base 2. every vertical line intersects it in at most one point. Equiv- ct Taking ± such that ² = c± log ± and noting that ®(G) ¸ alently, an x-monotone curve is the curve of a continuous n Â(G) for every graph G with n vertices, we have the following function de¯ned on an interval. A pair of convex sets or corollary of the previous theorem. x-monotone curves can have arbitrarily many intersection points between their boundaries. Theorem 5 and Theorem Corollary 2. For each ² > 0 and positive integer t, 7 below are similar to Theorem 1, but for intersection graphs there is ± = ±(²; t) > 0 such that if G is an intersection of convex sets and x-monotone curves, respectively. graph of a t-intersecting family of n curves in the plane, ± then G has a clique of size at least n or an independent set Theorem 5. If G is a Kk-free intersection graph of n 1¡² of size at least n .
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