The Number of Edges in k-Quasi-planar Graphs The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Fox, Jacob, János Pach, and Andrew Suk. “The Number of Edges in $k$-Quasi-planar Graphs.” SIAM Journal on Discrete Mathematics 27, no. 1 (March 19, 2013): 550-561. © 2013, Society for Industrial and Applied Mathematics As Published http://dx.doi.org/10.1137/110858586 Publisher Society for Industrial and Applied Mathematics Version Final published version Citable link http://hdl.handle.net/1721.1/79615 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. SIAM J. DISCRETE MATH. c 2013 Society for Industrial and Applied Mathematics Vol. 27, No. 1, pp. 550–561 THE NUMBER OF EDGES IN k-QUASI-PLANAR GRAPHS∗ † ´ ‡ † JACOB FOX ,JANOS PACH , AND ANDREW SUK Abstract. A graph drawn in the plane is called k-quasi-planar if it does not contain k pair- wise crossing edges. It has been conjectured for a long time that for every fixed k,themaximum number of edges of a k-quasi-planar graph with n vertices is O(n). The best known upper bound c n n O(log k) n n α(n) k is (log ) . In the present paper, we improve this bound to ( log )2 in the special α n case where the graph is drawn in such a way that every pair of edges meet at most once. Here ( ) denotes the (extremely slowly growing) inverse of the Ackermann function. We also make further progress on the conjecture for k-quasi-planar graphs in which every edge is drawn as an x-monotone curve. Extending some ideas of Valtr, we prove that the maximum number of edges of such graphs ck6 is at most 2 n log n. Key words. topological graphs, quasi-planar graphs, Turan-type problems AMS subject classifications. 05C35, 05C10, 68R10, 52C10 DOI. 10.1137/110858586 1. Introduction. A topological graph is a graph drawn in the plane such that its vertices are represented by points and its edges are represented by nonself-intersecting arcs connecting the corresponding points. In notation and terminology, we make no distinction between the vertices and edges of a graph and the points and arcs repre- senting them, respectively. No edge is allowed to pass through any point representing a vertex other than its endpoints. Any two edges can intersect only in a finite number of points. Tangencies between the edges are not allowed. That is, if two edges share an interior point, then they must properly cross at this point. A topological graph is simple if every pair of its edges intersect at most once: at a common endpoint or at a proper crossing. If the edges of a graph are drawn as straight-line segments, then the graph is called geometric. Finding the maximum number of edges in a topological graph with a forbidden crossing pattern is a fundamental problem in extremal topological graph theory (see [2,3,4,6,10,12,16,21,23]).ItfollowsfromEuler’spolyhedralformulathatevery topological graph on n vertices and with no two crossing edges has at most 3n − 6 edges. A graph is called k-quasi-planar if it can be drawn as a topological graph with no k pairwise crossing edges. A graph is 2-quasi-planar if and only if it is planar. According to an old conjecture (see Problem 1 in section 9.6 of [5]), for any fixed k ≥ 2 there exists a constant ck such that every k-quasi-planar graph on n vertices has at most ckn edges. Agarwal et al. [4] were the first to prove this conjecture for ∗Received by the editors December 12, 2011; accepted for publication (in revised form) October 30, 2012; published electronically March 19, 2013. A preliminary version of this paper, with A. Suk as its sole author, appears in Proceedings of the 19th International Symposium on Graph Drawing (GD 2011, TU Eindhoven), Lecture Notes in Comput. Sci. 7034, Springer-Verlag, New York, 2011, pp. 266–277. http://www.siam.org/journals/sidma/27-1/85858.html †Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02114 ([email protected], [email protected]). The first author was supported by a Simons Fellowship and NSF grant DMS 1069197. The third author was supported by an NSF Postdoctoral Fellowship. ‡EPFL Lausanne, CH-1015 Lausanne, Switzerland and Courant Institute, New York University, Downloaded 07/17/13 to 18.51.1.228. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php New York, NY 10012 ([email protected]). This author was supported by Hungarian Science Foun- dation EuroGIGA grant OTKA NN 102029, by Swiss National Science Foundation grant 200021- 125287/1, and by NSF grant CCF-08-30272. 550 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE NUMBER OF EDGES IN k-QUASI-PLANAR GRAPHS 551 simple 3-quasi-planar graphs. Later, Pach, Radoiˇci´c, and T´oth [17] generalized the result to all 3-quasi-planar graphs. Ackerman [1] proved the conjecture for k =4. For larger values of k, first Pach, Shahrokhi, and Szegedy [18] showed that every 2k−4 simple k-quasi-planar graph on n vertices has at most ckn(log n) edges. For k ≥ 3 and for all (not necessarily simple) k-quasi-planar graphs, Pach, Radoiˇci´c, and 4k− 12 T´oth [17] established the upper bound ckn(log n) . Plugging into these proofs ≥ the above mentioned result of Ackerman [1], for k 4, we obtain the slightly better 2k−8 4k−16 bounds ckn(log n) and ckn(log n) , respectively. For large values of k,the exponent of the polylogarithmic factor in these bounds was improved by Fox and Pach [10], who showed that the maximum number of edges of a k-quasi-planar graph O(log k) on n vertices is n(log n) . For the number of edges of geometric graphs, that is, graphs drawn by straight- line edges, Valtr [22] proved the upper bound O(n log n ). He also extended this result to simple topological graphs whose edges are drawn as x-monotone curves [23]. The aim of this paper is to improve the best known bound, n(log n)O(log k),onthe number of edges of a k-quasi-planar graph in two special cases: for simple topological graphs and for not necessarily simple topological graphs whose edges are drawn as x-monotone curves. In both cases, we improve the exponent of the polylogarithmic factor from O(log k)to1+o(1). Theorem 1.1. Let G =(V,E) be a k-quasi-planar simple topological graph with α n ck n vertices. Then |E(G)|≤(n log n)2 ( ) ,whereα(n) denotes the inverse of the Ackermann function and ck is a constant that depends only on k. Recall that the Ackermann (more precisely, the Ackermann–P´ eter) function A(n) is defined as follows. Let A1(n)=2n, and let Ak(n )=Ak−1(Ak(n − 1)) for k = n 2, 3,.... In particular, we have A2(n)=2 ,andA3(n) is an exponential tower of n { ≥ ≥ two’s. Now let A(n)=An(n), and let α(n) be defined as α(n)=min k 1:A(k) n}. This function grows much slower than the inverse of any primitive recursive function. Theorem 1.2. Let G =(V,E) be a k-quasi-planar (not necessarily simple) topological graph with n vertices, whose edges are drawn as x-monotone curves. Then | |≤ ck6 E(G) 2 n log n,wherec is an absolute constant. In both proofs, we follow the approach of Valtr [23] and apply results on gener- alized Davenport–Schinzel sequences. An important new ingredient of the proof of Theorem 1.1 is a corollary of a separator theorem established in [9] and developed in [8]. Theorem 1.2 is not only more general than Valtr’s result, which applies only to simple topological graphs, but its proof gives a somewhat better upper bound: we use a result of Pettie [20], which improves the dependence on k from double exponential to single exponential. 2. Generalized Davenport–Schinzel sequences. The sequence u = a ,a , 1 2 ...,am is called l-regular if any l consecutive terms are pairwise different. For integers l, t ≥ 2, the sequence S = s1,s2,...,slt of length lt is said to be of type up(l, t) if the first l terms are pairwise different and si = si+l = si+2l = ···= si+( t−1)l Downloaded 07/17/13 to 18.51.1.228. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php for every i,1≤ i ≤ l. For example, a, b, c, a, b, c, a, b, c, a, b, c Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 552 JACOB FOX, JANOS´ PACH, AND ANDREW SUK is a type up(3, 4) sequence or, in short, an up(3, 4) sequence. We need the following theorem of Klazar [13] on generalized Davenport–Schinzel sequences. Theorem 2.1 (Klazar). For l ≥ 2 and t ≥ 3, the length of any l-regular sequence over an n-element alphabet that does not contain a subsequence of type up(l, t) has length at most · (lt−3) · 10α(n)lt n l2 (10l) . ≥ For l 2, the sequence S = s1,s2,...,s3l−2 of length 3l − 2issaidtobeoftype up-down-up(l) if the first l terms are pairwise different and si = s2l−i = s(2l−2)+i ≤ ≤ for every i, 1 i l.