Andres J. Ruiz-Vargas

Topological graphs: empty triangles and disjoint matchings

A simple topological graph is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once. We present a novel tool for finding crossing free subgraphs in simple topological graphs. Using this tool, we solve the following two problems. Let G be a complete simple topological graph on n vertices. The three edges induced by any triplet of vertices in G form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an empty triangle. In 1998, Harborth proved that G has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least 2n/3. We settle Harborths conjecture in the affirmative. We also present a new proof of a result by Suk stating that every complete simple topological graph on n vertices contains Ω(n1/3) pairwise disjoint edges.

On the page number of upward planar directed acyclic graphs

In this paper we study the page number of upward planar directed acyclic graphs. We prove that: (1) the page number of any n-vertex upward planar triangulation G whose every maximal 4-connected component has page number k is at most min {O(klogn),O(2k)}; (2) every upward planar triangulation G with

Disjoint edges in topological graphs and the tangled-thrackle conjecture

o(\frac{n}{\log n})

Cross-sections of line configurations in R3 and ( d − 2)-flat configurations in Rd

diameter has o(n) page number; and (3) every upward planar triangulation has a vertex ordering with o(n) page number if and only if every upward planar triangulation whose maximum degree is

Many disjoint edges in topological graphs

O(\sqrt n)

Empty Triangles in Complete Topological Graphs

does.

Disjoint Edges in Topological Graphs and the Tangled-Thrackle Conjecture

It is shown that for a constant t ? N , every simple topological graph on n vertices has O ( n ) edges if the graph has no two sets of t edges such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is K t , t -free). As an application, we settle the tangled-thrackle conjecture formulated by Pach, Radoicic, and Toth: Every n -vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most O ( n ) edges.

Empty triangles in complete topological graphs

Abstract We consider sets L = { l 1 , … , l n } of n labeled lines in general position in R 3 , and study the order types of point sets { p 1 , … , p n } that stem from the intersections of the lines in L with (directed) planes Π, not parallel to any line of L , that is, the proper cross-sections of L . As two main results, we show that the number of different order types that can be obtained as cross-sections of L is O ( n 9 ) when considering all possible planes Π, and O ( n 3 ) when restricting considerations to sets of pairwise parallel planes, where both bounds are tight. The result for parallel planes implies that any set of n points in R 2 moving with constant (but possibly different) speeds along straight lines forms at most O ( n 3 ) different order types over time. We further generalize the setting from R 3 to R d with d > 3 , showing that the number of order types that can be obtained as cross-sections of a set of n labeled ( d − 2 ) -flats in R d with planes is O ( ( ( n 3 ) + n d ( d − 2 ) ) ) .

Tangles and degenerate tangles

A monotone cylindrical graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called simple if any pair of its edges have at most one point in common: an endpoint or a point at which they properly cross. We say that two edges are disjoint if they do not intersect. We show that every simple complete monotone cylindrical graph on n vertices contains ( n 1 ) pairwise disjoint edges for any 0 . As a consequence, we show that every simple complete topological graph (drawn in the plane) with n vertices contains ( n 1 2 ) pairwise disjoint edges for any 0 . This improves the previous lower bound of ( n 1 3 ) by Suk which was reproved by Fulek and Ruiz-Vargas. We remark that our proof implies a polynomial time algorithm for finding this set of pairwise disjoint edges.

On the Page Number of Upward Planar Directed Acyclic Graphs

A simple topological graph is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once. Let