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

Coloring Kk-free intersection graphs of geometric objects in the plane

∗ † 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 χ(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 ∈ C are joined by an parameters is a notoriously difficult 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 first 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 ε > 0 and lies of curves in the plane. Following [41], some algorithmic for every positive integer t, there exist δ > 0 and a positive aspects of this approach were explored in [4]. Obviously, α(G) ≥ n/χ(G) holds for every graph G with n vertices. integer n0 such that every topological graph with n ≥ n0 vertices, at least n1+ε edges, and no pair of edges intersecting Therefore, any upper bound on the chromatic number yields in more than t points, has at least nδ 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 α(G) is the size of Asplund and Grunbaum¨ [7] almost half a century ago. Since the largest independent set, the clique number ω(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. † 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 first 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 χ(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 finite boolean combination of polynomial equations terval I ⊂ 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 κ such that there is a points. representation of S with dimension d at most κ, the number The following theorem gives an upper bound on the chro- of equations and inequalities at most κ, and each of them matic number of the intersection graph of any t-intersecting has degree at most κ. (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. χ(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 description 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 χ(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 δ such that ² = cδ log δ and noting that α(G) ≥ alently, an x-monotone curve is the curve of a continuous n χ(G) for every graph G with n vertices, we have the following function defined 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 δ = δ(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, δ 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 . convex sets in the plane, then A Jordan region is a subset of the plane that is homeo- log n 13 log k χ(G) ≤ c , morphic to a closed disk. We say that a Jordan region α log k contains another Jordan region β if β lies in the interior of α. Define an r-region to be a subset of the plane that is the where c is an absolute constant. union of at most r Jordan regions. Call these (at most r) c Jordan regions of an r-region the components of the r-region. Taking δ such that ² = 13δ log δ and noting that α(G) ≥ n A crossing between a pair of Jordan regions is either a χ(G) for every graph G with n vertices, we have the following crossing between their boundaries or a containment between corollary of the previous theorem. them. A family of Jordan regions is t-intersecting if the boundaries of any two of them intersect in at most t points. Corollary 6. For each ² > 0 there is δ = δ(²) > 0 such A family of r-regions is t-intersecting if the family of all of that every intersection graph of n convex sets in the plane their components is t-intersecting. has a clique of size at least nδ or an independent set of size By slightly fattening curves in the plane, it is easy to see at least n1−². that if G is an intersection graph of a t-intersecting family of curves, then G is also an intersection graph of a 4t- A result of a similar flavor was obtained by Larman et al. intersecting family of Jordan regions. Theorem 1 and its [33]. They showed that for every positive integer k, every proof generalize in a straightforward manner to intersection family of n convex sets in the plane has an independent set of graphs of t-intersecting families of Jordan regions. With a size k or a clique of size at least n/k4. Notice that Corollary little more effort, we will generalize Theorem 1 to intersec- 6 only applies in the case that the clique number is not too tion graphs of t-intersecting family of r-regions. large while the result of Larman et al. [33] only applies when the independence number is not too large. Theorem 7. If G is a Kk-free intersection graph of n settles this conjecture and generalizes the result of Aronov x-monotone curves in the plane, then et al. [6]. 15 log k χ(G) ≤ (c log n) , Theorem 9. For every ² > 0 and every integer t > 0, where c is an absolute constant. there exist δ > 0 and a positive integer n0 with the following property. If G is a topological graph with n ≥ n0 vertices In Theorem 5 and Theorem 7, the constant factors in the and at least n1+² edges such that no pair of them intersect exponent can be improved by more careful calculation. in more than t points, then G has nδ pairwise crossing edges.

1.2 Applications to topological graphs Notice that every lower bound on the independence num- We next discuss a few applications of the above results to ber (and, hence, every upper bound on the chromatic num- graph drawings, beginning with some pertinent background. ber) of intersection graphs of curves yields an upper bound A topological graph is a graph drawn in the plane so that its on the number of edges of a topological graph. To see this, vertices are represented by points and its edges are repre- consider a topological graph G with n vertices. Delete from sented by curves connecting the corresponding points such each edge a small neighborhood around its endpoints, and that no curve passes through a point representing a vertex take the intersection graph G0 of the resulting curves. Any different from its endpoints. A topological graph is simple if independent set in G0 corresponds to a planar subgraph of any pair of its edges have at most one point in common. A G, so that the independence number of G0 is at most 3n − 6. geometric graph is a (simple) topological graph whose edges Therefore, Theorem 8 follows from Theorem 1 and Theorem are represented by straight-line segments. 9 follows from Corollary 2. In the same way the conjecture It follows by a simple application of Euler’s Polyhedral that the maximum number of edges of a topological graph Formula that every planar graph of n vertices has at most with n vertices and no k pairwise crossing edges is Ok(n) 3n − 6 edges. A topological graph is called k-quasi-planar if would be a direct consequence of the following general con- no k edges pairwise intersect. In particular, a 2-quasi-planar jecture. graph is just a planar graph. According to an old conjecture (see, e.g., Problem 6 in [38]), for any positive integer k, there Conjecture 10. For every positive integer k, there is is a constant Ck such that every k-quasi-planar topological ck > 0 such that every Kk-free intersection graph of curves graph on n vertices has at most Ckn edges. In the case in the plane has an independent set of size ckn. k = 3, Agarwal et al. [2] proved the conjecture for simple topological graphs. Later Pach, Radoiˇcić, and G. Tóth [39] These results suggest that the extra restriction that curves extended the result for all topological graphs. More recently, connect vertices of a graph may be unnecessary for many of Ackerman [1] proved the conjecture for k = 4. the problems in geometric graph theory. There also has been progress in the general case. Pach, The following result improves the exponent in the poly- Shahrokhi, and Szegedy [40] proved that every k-quasi-planar logarithmic factor in the upper bound for topological graphs simple topological graph on n vertices has at most from O(k) to O(log k). 2k−4 ckn(log n) edges. Plugging into the proof the result of Agarwal et al. [2], this upper bound can be improved to Theorem 11. Every k-quasi-planar topological graph with 2k−6 c log k ckn(log n) . Analogously, using the result of Ackerman n vertices has at most n (log n) edges, where c is an ab- 2k−8 [1] instead, we obtain ckn(log n) . Valtr [48] proved that solute constant. every k-quasi-planar geometric graph on n vertices has at We have the following immediate corollary. most ckn log n edges. In [49], he extended this result to topological graphs with edges drawn as x-monotone curves. Pach, Radoiˇcić, and G. Tóth [39] proved that every k-quasi- Corollary 12. For each ² > 0 there is δ > 0 and n0 such that every topological graph with n ≥ n0 vertices and planar topological graph with n vertices has at most 1+² δ/ log log n 4k−12 at least n edges has n pairwise crossing edges. ckn(log n) edges, and by the result of Ackerman [1], 4k−16 this can be improved to ckn(log n) . The following theorem improves the exponent in the poly- A string graph is an intersection graph of curves in the logarithmic factor from O(k) to O(log k) for simple topolog- plane. An incomparability graph of a partially ordered set P ical graphs. has vertex set P and two elements of P are adjacent if and only if they are incomparable in P . The proof of Theorem Theorem 8. Every k-quasi-planar topological graph with 11 uses a recent result of the authors showing that string graphs and incomparability graphs are closely related. n vertices and no pair of edges intersecting in more than c log k In Section 2, we prove Theorem 1 and Theorem 3. In t points has at most n c log n edges, where c is an t log k Section 3, we establish a separator theorem which is used in absolute constant and ct only depends on t. the proof of Theorems 5 and 7. In Section 4, we establish Theorems 5 and 7. In Section 5 we prove Theorem 11. It was shown in [6] that everyp complete geometric graph on n vertices contains at least n/12 pairwise crossing edges. It was noted in [42] that the result of Pach, Shahrokhi, and 2. PROOFS OF THEOREM 1 AND Szegedy [40] implies that every complete simple topological THEOREM 3 log n graph has at least c log log n pairwise crossing edges. Pach The proof of Theorem 1 uses a separator theorem due and G. Tóth [42] conjectured that there is δ > 0 such that to the authors [12] (see Corollary 14 below) and a Turán- every complete simple topological graph on n ≥ 5 vertices type theorem from [15] on intersection graphs of curves (see has at least nδ pairwise crossing edges. Our next theorem Lemma 15). A separator for a graph G = (V,E) is a subset V0 ⊂ independent set of vertices in G and color its elements with V such that there is a partition V = V0 ∪ V1 ∪ V2 with the first color. Then pick a maximum independent set from 2 |V1|, |V2| ≤ 3 |V | and no vertex in V1 is adjacent to any vertex the graph induced by the uncolored vertices and color its in V2. The well-known separator theorem by Lipton and elements with the second color, and continue picking out Tarjan [34] states that every√ planar graph with n vertices maximum independent sets from the remaining uncolored has a separator of size O( n). By a beautiful theorem of vertices until all vertices are colored. Koebe [28], every planar graph can be represented as the We first give an upper bound on the number of colors used intersection graph of closed disks in the plane with disjoint to color half of the vertices of G. Each of the color classes interiors. Miller, Teng, Thurston, and Vavasis [37] found used to color the first half of the vertices of G has size at least a generalization of the Lipton-Tarjan separator theorem to αF (n/2). Hence, the number of colors used in coloring half higher dimensions. They proved that the intersection graph of the vertices of G is at most d n/2 e ≤ d n e, where d αF (n/2) αF (n) of any family of n balls in R such that no k of them have the inequality follows from subadditivity of α . Therefore, 1/d 1−1/d F a point in common has a separator of size O(dk n ). to color all but at most n/2k vertices of G, we use at most P (See also [46].) k−1 n/2j n d j e ≤ kd e colors. Taking k = log n, we Fox and Pach [12] established the following generalization j=0 αF (n/2 ) αF (n) of the separator theorems of Lipton and Tarjan and of Miller can properly color all vertices using at most dn/αF (n)e log n et al. [37] in two dimensions. colors. By Lemma 16, to establish Theorem 1, it suffices to prove Theorem 13. [12] If C is a finite family of Jordan regions the following result. with a total of m crossings,√ then the intersection graph of C has a separator of size O( m). Theorem 17. If G = (V,E) is a Kk-free intersection graph of a t-intersecting family of n curves in the plane, The following result is a corollary of Theorem 13. then log n −c log k Corollary 14. [12] If C is a finite family of curves in α(G) ≥ n c , t log k the plane with a total of m crossings,√ then the intersection graph of C has a separator of size O( m). where c is an absolute constant and ct only depends on t.

Proof. Let S0 = {V } be the family consisting of a single The constant in the big-O notation in both Theorem 13 set, V . At step i (i = 1, 2,...), we replace each W ∈ Si−1 and Corollary 14 can easily be taken to be 100, though a satisfying |W | ≥ 2 by either one or two subsets of W such detailed analysis of the proof gives a much better constant. that the resulting family Si consists of pairwise disjoint sub- A bi-clique is a complete bipartite graph whose two parts sets of V and no edge of G connects two vertices belonging differ in size by at most one. The following theorem is the 0 00 to distinct membersW ,W ∈ Si. We proceed as follows. second main tool in the proof of Theorem 1. 3 −8 −1 log k Let ² = 10 t log n . If the subgraph of G induced 2 Lemma 15. [15] Every intersection graph of n curves in by W ∈ Si−1 has at least ²|W | edges, then apply Lemma 15 2 the plane with at least ²n edges and no pair of curves in- to obtain disjoint subsets W1 and W2 with |W1| = |W2| ≥ c tersecting in more than t points contains a bi-clique of size ct² |W | such that every vertex in W1 is adjacent to every c at least ct² n, where c is an absolute constant and ct > 0 vertex in W2. We may assume without loss of generality depends only on t. that the clique number of the subgraph of G induced by W1 is at most the clique number of the subgraph induced by A family of graphs is said to be hereditary if it is closed by W2, so that the clique number of the subgraph induced by taking induced subgraphs. A family of graphs F is normal if W1 is at most half of the clique number of the subgraph of every graph G ∈ F is a proper induced subgraph of another 0 G induced by W . In this case, in Si we replace W by W1. graph G ∈ F. For any family F of graphs, let αF (n) = If the subgraph of G induced by W has fewer than ²|W |2 minG∈F,v(G)=n α(G) and let χF (n) = maxG∈F,v(G)=n χ(G). edges, then apply Corollary 14 to obtain two disjoint subsets For example, if F is a hereditary family and for every integer W1,W2 ⊂ W such that n there is a graph in F with clique number at least n, then p √ 2 αF (n) = 1 and χF (n) = n. If F is a hereditary normal |W | − |W1| − |W2| ≤ 100 t²|W | = 100 t²|W |, family, then it is easy to show that αF and χF are increasing, n |W1|, |W2| ≤ 2|W |/3, and no vertex in W1 is adjacent to any subadditive functions of n. Clearly, we have αF (n) ≥ , χF (n) vertex in W2. In this case, we replace W ∈ Si−1 by W1 and as α(G) ≥ n holds for every graph G with n vertices. The χ(G) W2. following lemma essentially shows that the last inequality Following this procedure, we build a tree of subsets of V , is tight apart from a logarithmic factor, that is, n log n is χF (n) with V being its root, so that the vertices of the tree at roughly an upper bound on αF (n). More precisely, we have height i are the members of Si. Any vertex W of the tree the following lemma. has either one or two children, which are subsets of W . Any path in this tree connecting the root V to a leaf has fewer 2 Lemma 16. If F is a hereditary normal family of graphs, than log2 k nodes W with at least ²|W | edges; each of these n then for all n, χF (n) ≤ d edlog ne. nodes has precisely one child. Since the size of a child Wi is αF (n) at most 2/3 times the size of its parent W , the height of the Proof. Let G ∈ F with n vertices. For simplicity, we i tree is at most log3/2 n. Therefore, the graph G must have will assume that n = 2 is a perfect power of 2, although an independent set of size at least the proof works as well for n not a power of 2. The proof 1/2 1/2 log3/2 n c log k is by a straightforward greedy algorithm: take a maximum (1 − 100t ² ) (ct² ) n 1/2 1/2 −100t ² log3/2 n c log k ≥ 4 (ct² ) n h/r. The induced subgraph of at least one of the two vertex classes of this bi-clique is K -free. In this case, with dk/2e 3c log k a different value of c, the first of the three inequalities is −1/10 −3 −1 1/3 log k ≥ k 10 (t ct) n, satisfied. log n Case 2: ² < δ. By Theorem 13, there are disjoint sub- where the first inequality uses the fact that 1 − x ≥ 4−x 2 families C1, C2 of C with |C1|, |C2| ≤ 3 n, holds for 0 ≤ x ≤ 1/2. This completes the proof, noting √ 2 2 1/2 1/2 that for Theorem 17, we have to pick ct and c different from |C| − |C1| − |C2| ≤ 100 ²r tn < 100δ rt n, the constants ct and c that we used from Lemma 15. and no Jordan region in C1 intersects any Jordan region in C . For 0 ≤ i ≤ r, let V ⊂ V consist of all those r-regions in The proof of Theorem 3 is a variant of the above argu- 2 i V that have all of their componentsS in C1 ∪ C2 and exactly i ment. We need the following straightforward generalization r components in C . Note that |V \ V | < 100δ1/2rt1/2n. of Lemma 15. 1 i=0 i Case 2a: There is i ∈ {1, . . . , r − 1} such that |Vi| ≥ 1/2 1/2 Lemma 18. [15] The intersection graph of any t-intersecting 100δ t n. In this case, the components of Vi in C1 form 2 a t-intersecting family of i-regions. So there is a subfamily family of n Jordan regions with at least ²n edges contains 0 c Vi ⊂ Vi of n1 := It(|V1|, k, i) r-regions such that no pair a bi-clique of size at least ct² n, where c is an absolute con- of them have intersecting components in C1. Furthermore, stant and ct > 0 depends only on t. 00 0 there exists a subfamily Vi ⊂ Vi of It(n1, k, r − i) r-regions By Lemma 16, to prove Theorem 3, it suffices to establish in Vi such that no pair of them have intersecting components the following result. in C2. Hence, these r-regions form an independent set of size It(n1, k, r − i) in the intersection graph. 1/2 Theorem 19. If G = (V,E) is a Kk-free intersection Case 2b: |Vi| < 100δt n for i ∈ {1, . . . , r − 1}. Since 2 graph of a t-intersecting family of n r-regions, then |Ci| ≤ 3 |C| for i ∈ {1, 2}, then |V0| and |Vr| each have car- 1 −cr log k dinality at most (1 − 3r )n. Notice that every r-region in log n V0 is disjoint from every r-region in Vr and |V0| + |Vr| ≥ α(G) ≥ n ct,r , 1/2 1/2 log k n − 200δ t rn. Letting a = |V0| and b = |Vr|, we obtain an independent set of size at least It(a, k, r) + It(b, k, r). where c is an absolute constant and ct,r depends only on t and r. −8 −1 −4 log k 3 Fixing δ = 10 t r ( log n ) , applying triple induction on n, k, and r, using Lemma 20, we arrive at Theorem 19, Let It(n, k, r) be the minimum of α(G) taken over every and hence also at Theorem 3. Kk-free graph G that is an intersection graph of a t-intersecting family of at least n r-regions. The proof of Theorem 19, 3. A SEPARATOR THEOREM FOR which gives a lower bound on It(n, k, r), is by triple induction on n, k, and r. The proof of the nontrivial base case OUTERSTRING GRAPHS r = 1 is essentially identical to the proof of Theorem 1, ex- A family C of curves in the plane is said to be grounded if cept that we use Lemma 18 instead of Lemma 15 and Theo- there is a closed (Jordan) curve γ such that every member rem 13 instead of Corollary 14. The other base cases, which of C has one endpoint on γ and the rest of the curve lies in are trivial, are when n = 1 (in which case It(1, k, r) = 1), the exterior of γ. The intersection graph of a collection of and when k = 2 (in which case It(n, 2, r) = n). The induc- grounded curves is called an outerstring graph. tion is then straightforward, using the following lemma. The members of a family C of n grounded curves can be cyclically labeled in a natural way, according to the order Lemma 20. For every positive integer t, there are con- of the endpoints of the curves along the ground γ. Start by stants ct > 0 and c such that the following is true. For any assigning the label 0 to any member of C, and then proceed δ > 0 and for any positive integers n, k, r, at least one of the to label the curves clockwise, breaking ties arbitrarily, so following three inequalities hold: that the (i + 1)-st member of C has label i ∈ Z. Define the

−c c distance d(i, j) between a pair of grounded curves in C as 1. It(n, k, r) ≥ It(ctr δ n, dk/2e, r). the cyclic distance between their labels i, j ∈ Z. That is, let

2. It(n, k, r) ≥ It(a, k, r) + It(b, k, r) where d(i, j) := min(|i − j|, n − |i − j|). a + b ≥ n − 200δ1/2rt1/2n and a, b ≤ (1 − 1 )n. 3r Let [i, j] denote the cyclic interval of elements {i, i+1, . . . , j}.

3. It(n, k, r) ≥ It(n1, k, i) where n1 = It(n2, k, r − i), In this section, we prove the following separator theorem 1/2 1/2 for outerstring graphs. We then show that this result is best 1 ≤ i ≤ r − 1, and n2 = d100δ t ne. possible apart from the constant factor. In the next section, Proof. Let G = (V,E) be a Kk-free intersection graph we will use this separator theorem to prove Theorems 5 and of a t-intersecting family of n r-regions with independence 7. 2 number α(G) = It(n, k, r). Let ²n be the number of edges of G. Let C denote the family of all the components, so Theorem 21. Every outerstring graph with m edges and |C| ≤ rn. maximum√ degree ∆ has a separator of size at most 4 min(∆, m). Case 1: ² ≥ δ. The family C has at most rn Jordan re- 2 gions and at least ²n intersecting pairs, so applying Lemma Notice that the upper bound√ on the size of the separator is 18, the intersection graph G(C) contains a bi-clique of size the minimum of 4∆ and 4 m. We ﬁrst prove the following 2 c h ≥ ct(²/r ) n. Then G contains a bi-clique of size at least lemma, and then deduce Theorem 21. a n n of edges of the outerstring graph. A split graph is a graph ≥ 3 < 3 a b whose vertex set can be partitioned into a clique and an n ≥ 3 independent set. It is an easy exercise to show that every e n c split graph is an outerstring graph. It is also straightforward ≥ 3 n < 3 d to check that for each ² > 0, there is C² such that if C²n ≤ b f m ≤ n2/18, then there is a split graph G with n vertices L L and at most m edges√ such that every separator for G has size at least (1 − ²) 2m. Indeed, the desired split graph

Figure 1: On the left: there are two curves, Ca and can be chosen, with high probability, to be the random split Cb, whose cyclic distance along L is at least n/3. On graph whose vertex set V has n vertices and a partition the right: the maximum distance between any two V = A ∪ B into√ a clique A and an independent set B with arcs is less than n/3. |A| = d(1 − ²/2) 2me, and for each vertex v√∈ A, the set of neighbors of v in B is a random subset of d² m/4e vertices from B. It is also easy to check that for ∆ even and at most n/3, the Cayley graph with vertex set Zn and two vertices Lemma 22. Every outerstring graph with maximum de- are adjacent if and only if they have cyclic distance at most gree ∆ has a separator of size at most 4∆. ∆/2 is an outerstring graph and its smallest separator has Proof. Let G be the outerstring graph of a collection size ∆. These two constructions demonstrate that Theorem 21 is tight up to the constant factor. C = {C0,...,Cn−1} of grounded curves, with this cyclic labeling. We may assume without loss of generality that Recall that the incomparability graph G = GP,< of a par- every curve intersects at least one other curve, that is, G has tially ordered set (P, <) is the graph with vertex set P in which two vertices are connected by an edge if and only if no isolated vertices. Let (Ca,Cb) be a pair of intersecting curves whose distance is maximum. they are incomparable by the relation <. Since every incomparability graph is an outerstring graph (see [20], [43], [45]), Case 1: d(a, b) ≥ n ; see the left-hand side of Figure 1. 3 it follows that every incomparability graph G with maxi- Let V be the set of curves that intersect at least one of the 0 mum degree ∆ and m edges has a separator of size at most curves C or C . Since C and C each intersect at most ∆ √ a b a b 4 min(∆, m). This can easily be improved upon as the sep- other curves, V has at most 2∆ elements. Let V consist of 0 1 arator size in an incomparability graph can be bounded by all curves in C\ V whose labels belong to the cyclic interval 0 a constant times the average degree. [a, b], and let V2 consist of all curves in C\ V0 whose labels belong to [b, a]. Notice that |V | ≤ 2∆, |V |, |V | ≤ 2 |C|, and 0 1 2 3 Proposition 23. Every incomparability graph with n ver- no curve in V1 intersects any curve in V2. Therefore, G has tices and m edges has a separator of size at most 6m/n. a separator of size at most 2∆. n Proof. Let G = GP,< be an incomparability graph with Case 2: d(a, b) < 3 ; see the right-hand side of Figure n m edges. Pick any linear extension of the partial order <, 1. Let c ∈ Zn be defined by c ≡ b + d e (mod n). By 3 and let W ⊂ P denote the set of vertices belonging to the assumption, the curve Cc intersects at least one other curve middle third of P with respect to this linear ordering. Let v in C. Let d be a label such that Cd intersects Cc and d(c, d) is as large as possible. Finally, let e ∈ [b, c] and f ∈ [d, a] be an element of W whose neighborhood N(v) is as small as possible. Obviously, the set N(v) is a separator for G, and be two labels such that Ce intersects Cf and d(e, f) is as it has at most 6m/n elements. large as possible. Let V0 denote the set of curves in C that intersect at least one of the curves Ca, Cb, Ce, or Cf . Since each member of C intersects at most ∆ other curves, V0 4. PROOFS OF THEOREM 5 AND has at most 4∆ elements. Let V1 consist of all curves in THEOREM 7 C\ V0 whose labels belong to the cyclic interval [a, b], let V2 The following analogues of Lemma 15 for intersection graphs consist of all curves in C\ V0 whose labels belong to [b, e], of convex sets and x-monotone curves in the plane were ob- let V3 consist of all curves in C\ V0 whose labels belong to tained by the authors and Cs. Tóth in [16]. [e, f], and let V4 consist of all curves in C\ V0 whose labels belong to [f, a]. Clearly, we have |V | ≤ 4∆, |V | ≤ 2 |C| for 0 i 3 Lemma 24. [16] There is a constant c > 0 such that every i ∈ {1, 2, 3, 4}, and no curve in V intersects any curve in V i j intersection graph of n convex sets in the plane with at least with 1 ≤ i < j ≤ 4. Therefore, by combining the V ’s into i ²n2 edges contains a bi-clique with at least c²2n vertices in two sets each of cardinality at most 2|C|/3, the graph G has each of its vertex classes. a separator of size at most 4∆. Proof of Theorem 21: Let G be an outerstring graph with Lemma 25. [16] There is a constant c > 0 such that every 2 m edges. Delete vertices of G of maximum degree one by intersection graph of n x-monotone curves with at least ²n 2 0 edges contains a bi-clique with at least c ² n vertices one until the remaining√ induced subgraph G has maximum log 1/² log n degree at most ∆ := m/2. Let D denote the set of deleted√ in each of its vertex classes. vertices. The cardinality of D is at most m/∆ = 2 m. 0 0 By Lemma 22, G has a separator V0 of cardinality at most It was pointed out in [43], that a result of Fox [11] implies √ 0 4∆ = 2 m√. Then the set V0 := D∪V0 , which has cardinality that the dependence on n in Lemma 25 is tight. at most 4 m, is a separator for G. The proofs of Theorems 5 and 7 are so similar that we only We now discuss two constructions which together show include the proof of Theorem 5. For the proof of Theorem that apart from a constant factor of 4, Theorem 21 is best 5, we may assume that the convex sets are closed convex possible as a function of the maximum degree and number polygons. This is justified, since every intersection graph of finitely many convex sets in the plane is the intersection Putting together the last three lemmas, Theorem 5 fol- graph of convex closed polygons, as observed in [16]. lows. Let X(n, k) denote the maximum chromatic number over all Kk-free intersection graphs of n convex sets in the plane. 5. K-QUASI-PLANAR TOPOLOGICAL Let V (n, k) denote the maximum chromatic number over GRAPHS all Kk-free intersection graphs of n convex sets in the plane that each intersect the same vertical line L. In [20], it is shown that every incomparability graph is The following lemma relates V (n, k) and X(n, k). a string graph. Recently, the authors proved the following theorem which implies that every dense string graph contain Lemma 26. For all positive integers n and k, we have a dense subgraph which is an incomparability graph. n X(n, k) ≤ X(b c, k) + V (n, k). 2 Theorem 29. [13] There is a constant c1 such that for Proof. Let C be a family of n convex polygons in the every collection C of n curves in the plane whose intersection graph has ²|C|2 edges, we can pick for each curve γ ∈ C a plane, and let x1 ≤ ... ≤ xn denote the x-coordinates of 0 0 the leftmost points of the members of C. Let L be the ver- subcurve γ such that the intersection graph of {γ : γ ∈ C} has at least ²c1 |C|2 edges and is an incomparability graph. In tical line x = xd n e. Notice that every convex set whose 2 particular, every string graph on n vertices and ²n2 edges has rightmost point has x-coordinate smaller than x n is dis- d 2 e c1 2 joint from every convex set whose leftmost point has x- a subgraph with at least ² n edges that is an incomparability n graph. coordinate larger than x n . There are at most b c mem- d 2 e 2 bers of C whose rightmost points have x-coordinates smaller n Theorem 29 shows that string graphs and incomparabil- than xd n e and at most b c members whose leftmost points 2 2 ity graphs are closely related. The following result of the have x-coordinates larger than xd n e. Hence, all members 2 authors and Tóth shows that every dense incomparability of C that do not intersect L can be properly colored with n graph contains a large balanced complete bipartite graph. X(b 2 c, k) colors. The remaining members of C all intersect L, hence they can be colored with V (n, k) colors. Thus, we Lemma 30. [16] Every incomparability graph I with n n 2 have X(n, k) ≤ X(b 2 c, k) + V (n, k). vertices and ²n edges contains the complete bipartite graph K with t ≥ c ² n , where c is a positive absolute By iterating Lemma 26, we obtain t,t 2 log 1/² log n 2 constant. blogXnc n X(n, k) ≤ V (b c, k) ≤ (1 + log n)V (n, k). >From Theorem 29 and Lemma 30, we have the following 2i 2 i=0 corollary.

Let G(n, k) denote the maximum chromatic number over Corollary 31. Every string graph with n vertices and all Kk-free intersection graphs of n convex sets in the plane 2 ²n edges contains the complete bipartite graph Kt,t with that intersect a vertical line L and lie in the half-plane to c n t ≥ ² 3 , where c is an absolute constant. the right of L. log n 3 Lemma 27. For all positive integers n and k, we have The bisection width b(G) of a graph G = (V,E) is the least integer for which there is a partition V = V1 ∪ V2 such that V (n, k) ≤ G(n, k)2. 2 |V1|, |V2| ≤ 3 |V | and the number of edges between V1 and V2 Proof. Let C = {C1,...,Cn} be a family of convex poly- is b(G). The pair-crossing number pcr(G) of a graph G is the gons that intersect a vertical line L : x = x0. For 1 ≤ smallest number of pairs of edges that crossP in a drawing of G 2 i ≤ n, let Li denote the intersection of Ci with the left in the plane. For a graph G, let ssqd(G) = v∈V (G) deg(v) . half-plane {(x, y): x ≤ x0}, and let Ri denote the inter- We will use the following result of Kolman and Matouˇsek section of Ci with the right half-plane {(x, y): x ≥ x0}. [29]. Let L = {L1,...,Ln} and R = {R1,...,Rn}. Notice that the intersection graph of L can be properly colored with Lemma 32. [29] Every graph G on n vertices satisfies p p G(n, k) colors, and the intersection graph of R can be prop- b(G) ≤ c log n pcr(G) + ssqd(G) , erly colored with G(n, k) colors. Consider two proper color- 4 ings c : L → {1,...,G(n, k)} and c : R → {1,...,G(n, k)} 1 2 where c4 is an absolute constant. of the intersection graphs of L and R. Assigning each convex set Ci, 1 ≤ i ≤ n, the color (c1(Li), c2(Ri)), we obtain Proof of Theorem 11: Define T (n, k) to be the maximum a proper coloring of C with G(n, k)2 colors. Hence, we have number of edges in a k-quasi-planar topological graph with V (n, k) ≤ G(n, k)2. n vertices. We will prove by induction on n and k the upper bound Every intersection graph of convex sets in the plane with the property that all of them intersect a vertical line L and T (n, k + 1) ≤ n(log n)c5 log k lie in the half-plane to the right of L is an outerstring graph. where c is a sufficiently large absolute constant, which im- Therefore, to establish the following lemma, we may use 5 plies Theorem 11. the proof of Theorem 1, with the difference that Corollary Note that we have the simple bounds T (n, k) ≤ n , T (n, 1) = 14 and Lemma 15 have to be replaced by Theorem 21 and 2 0, and T (n, 2) = 3n − 6 for n ≥ 3. The last bound is from Lemma 24. the fact that every n-vertex planar graph has at most 3n−6 0 0 Lemma 28. There is a constant c > 0 such that G(n, k) ≤ edges. The induction hypothesis is that if n ≤ n and k ≤ k 0 log n 6 log k 0 0 0 0 0 0 c5 log k (c log k ) for all k and n with k ≤ n. and (n , k ) 6= (n, k), then T (n , k + 1) ≤ n (log n ) . 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