Complete Topological Graph Research Articles

Many disjoint edges in topological graphs

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 Ω(n1−ϵ) pairwise disjoint edges for any ϵ>0. As a consequence, we show that every simple complete topological graph (drawn in the plane) with n vertices contains Ω(n12−ϵ) pairwise disjoint edges for any ϵ>0. This improves the previous lower bound of Ω(n13) 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.

Many disjoint edges in topological graphs

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 Ω(n1−ϵ) pairwise disjoint edges for any ϵ>0. As a consequence, we show that every simple complete topological graph (drawn in the plane) with n vertices contains Ω(n12−ϵ) pairwise disjoint edges for any ϵ>0. This improves the previous lower bound of Ω(n13) 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.

Estimating the Number of Disjoint Edges in Simple Topological Graphs via Cylindrical Drawings

A topological graph drawn on a cylinder whose base is horizontal is angularly monotone if every vertical line intersects every edge at most once. Let $c(n)$ denote the maximum number $c$ such that every simple angularly monotone drawing of a complete graph on $n$ vertices contains at least $c$ pairwise disjoint edges. We show that for every simple complete topological graph $G$ there exists $\Delta$, $0<\Delta<n$, such that $G$ contains at least $\max \{\frac n\Delta, c(\Delta)\}$ pairwise disjoint edges. By combining our result with a result of Toth we obtain an alternative proof for the best known lower bound of $\Omega(n^\frac 13)$ on the maximum number of pairwise disjoint edges in a simple complete topological graph proved by Suk. Our proof is based on a result of Ruiz-Vargas.

Empty triangles in complete topological 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 G be a simple complete topological graph. The three edges induced by any triple 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 [Harborth, Heiko, Empty triangles in drawings of the complete graph, Discrete Mathematics, 191 (1998), 109–111; Brass, Peter, William O. J. Moser and János Pach, “Research Problems in Discrete Geometry” Springer, (2005)] proved that G has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least 2n/3. In this note, we verify Harborthʼs conjecture.

Improved Enumeration of Simple Topological Graphs

A simple topological graph T = (V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We generalize results of Pach and Toth and the author's previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph G with n vertices, m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize G is at most 2^O(n^2 log(m/n)), and at most 2^O(mn^{1/2} log n) if m < n^{3/2}. As a consequence we obtain a new upper bound 2^O(n^{3/2} log n) on the number of intersection graphs of n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with n vertices to 2^{n^2 alpha(n)^O(1)}, using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize G is at most 2^{m^2+O(mn)} and at least 2^Omega(m^2) for graphs with m > (6+epsilon)n.

Disjoint Edges in Complete Topological Graphs

It is shown that every complete $$n$$ -vertex simple topological graph has at $$\varOmega (n^{1/3})$$ pairwise disjoint edges, and these edges can be found in polynomial time. This proves a conjecture of Pach and Toth, which appears as Problem 5 from Chapter 9.5 in Research Problems in Discrete Geometry by Brass, Moser, and Pach.

Disjoint edges in topological graphs

A topological graph G is a graph drawn in the plane so that its edges are represented by Jordan arcs. G is called simple, if any two edges have at most one point in common. It is shown that the maximum number of edges of a simple topological graph with n vertices and no k pairwise disjoint edges is O(nlog4k−8n) edges. The assumption that G is simple cannot be dropped: for every n, there exists a complete topological graph of n vertices, whose any two edges cross at most twice.

Enumeration of simple complete topological graphs

A simple topological graph T = ( V ( T ) , E ( T ) ) is a drawing of a graph in the plane, where every two edges have at most one common point (an end-point or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We prove that the number of isomorphism classes of simple complete topological graphs on n vertices is 2 Θ ( n 4 ) . We also show that the number of weak isomorphism classes of simple complete topological graphs with n vertices and n 4 crossings is at least 2 n ( log n − O ( 1 ) ) , which improves the estimate of Harborth and Mengersen.

Enumeration of simple complete topological graphs

Abstract A simple topological graph T = ( V ( T ) , E ( T ) ) is a drawing of a graph in the plane, where every two edges have at most one common point (an end-point or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We prove that the number of isomorphism classes of simple complete topological graphs on n vertices is 2 Θ ( n 4 ) . We also show that the number weak isomorphism classes of simple complete topological graphs with n vertices and ( n 4 ) crossings is at least 2 n ( log n − O ( 1 ) ) , which improves the estimate of Harborth an Mengersen.

Unavoidable Configurations in Complete Topological Graphs

A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topological complete graph with n vertices has a canonical subgraph of size at least clog1/8n, which belongs to one of these classes. We also show that every complete topological graph with n vertices has a non-crossing subgraph isomorphic to any fixed tree with at most clog1/6n vertices.