Disjoint edges in complete topological graphs

Given a collection of objects C in the plane, the intersection graph G(C) has vertex set C and two objects are adjacent if and only if they have a nonempty intersection. The independence number of G(C) is the size of the largest independent set, that is, the size of the largest subfamily of pairwise disjoint objects in C. It is known that computing the independence number of an intersection graph is NP-hard [12], even for very simple objects such as rectangles and disks in the plane [3,11]. However, due to its applications in VLSI design [13], map labeling [2], and elsewhere, a lot of research has been devoted to developing polynomial-time approximation schemes (PTAS) for computing the independence number of intersection graphs (see [9] for more references). In this paper, we study the independence number of the intersection graph of edges in a complete simple topological graph.

A topological graph is a graph drawn in the plane such that its vertices are represented by points and its edges are represented by non-self-intersecting arcs connecting the corresponding points. The arcs are allowed to intersect, but they may not intersect vertices except for their endpoints. Furthermore, no three edges have a common interior point and no two edges are tangent, i.e., if two edges share an interior point, then they must properly cross at that point in common. A topological graph is simple if every pair of its edges intersect at most once. Two edges of a topological graph cross if their interiors share a point, and are disjoint if they neither share a common vertex nor cross. If the edges are drawn as straight-line segments, then the graph is geometric.

Let F denote the graph obtained from the complete graph on 5 vertices by subdividing each edge with an extra vertex. It is easy to see that every intersection graph G of curves in the plane does not contain F as an induced subgraph [6]. By applying a theorem of Erdős and Hajnal [7], every complete n-vertex simple topological graph contains e Ω( √ log n) edges that are either pairwise disjoint or pairwise crossing. However, it was suspected [16] that this bound is far from optimal. Fox and Pach [8] showed that there exists a constant δ > 0, such that every complete n-vertex simple topological graph contains Ω(n δ ) pairwise crossing edges. However, much weaker bounds were previously known for pairwise disjoint edges.

In 2003, Pach, Solymosi, and Tóth [16] showed that every complete n-vertex simple topological graph has at least Ω(log 1/6 n) pairwise disjoint edges. This lower bound was later improved by Pach and Tóth [17] to Ω(log n/ log log n). Recently, Fox and Sudakov [10] gave a modest improvement of Ω(log 1+ǫ n), where ǫ is a very small constant. We note that the previous two bounds hold for dense simple topological graphs. Pach and Tóth conjectured (see problem 5 of chapter 9.5 in [4]) that there exists a constant δ > 0 such that every complete n-vertex simple topological graph has at least Ω(n δ ) pairwise disjoint edges. Our main result settles the conjecture in the affirmative.

Theorem 1.1. Every complete n-vertex simple topological graph contains Ω(n 1/3 ) pairwise disjoint edges.

Note that Theorem 1.1 does not remain true if the simple condition is dropped. Indeed, in [17], Pach and Tóth gave a construction of a complete n-vertex topological graph such that every pair of edges intersect exactly once or twice.

In this section, we will recall one of the most useful parameters measuring the complexity of a set system: the dual shatter function. All of the following concepts and results can be found in Chapter 5 of [14]. Let (X, S) be a set system with ground set X, such that X is finite.

Definition 2.1. The dual shatter function of (X, S) is a function, denoted by π * S , whose value at m is defined as the maximum number of equivalence classes on X defined by an m-element subfamily Y ⊂ S, where two points x, y ∈ X are equivalent with respect to Y if x belongs to the same sets of Y as y does. In other words, π * S (m) is the maximum number of nonempty cells in the Venn diagram of m sets of S.

One of the main tools used to prove Theorem 1.1 is the following result of Chazelle and Welzl on matchings with low stabbing number. A similar approach was done by Pach in [15], who showed that every n-vertex complete geometric graph contains Ω(n 1/2 ) pairwise parallel edges, where two edges are parallel if they are the opposite sides of a convex quadrilateral. Given a set system (X, S) and a graph G = (X, E), we say that a set S ∈ S stabs edge uv ∈ E(G) if |S ∩ {u, v}| = 1. The stabbing number of G with respect to the set S is the number of edges of G stabbed by S, and the stabbing number of G is the maximum of stabbing numbers of G with respect to all sets of S.

). Let n be an even integer, and S be a set system on an n-point set X with π * S (m) ≤ Cm d for all m, where C and d are constants. Then there exists a perfect matching M on X (i.e. a set of n/2 vertex

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