On the Richter-Thomassen Conjecture about Pairwise Intersecting Closed Curves

A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any $n$ simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least $(1-o(1))n^2$. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class is touching every curve from the second class. (Two curves are said to be touching if they have precisely one point in common, at which they do not properly cross.) An important ingredient of our proofs is the following statement: Let $S$ be a family of the graphs of $n$ continuous real functions defined on $\mathbb{R}$, no three of which pass through the same point. If there are $nt$ pairs of touching curves in $S$, then the number of crossing points is $\Omega(nt\sqrt{\log t/\log\log t})$.Comment: To appear in SODA 201

Nested cycles in large triangulations and crossing-critical graphs

We show that every sufficiently large plane triangulation has a large collection of nested cycles that either are pairwise disjoint, or pairwise intersect in exactly one vertex, or pairwise intersect in exactly two vertices. We apply this result to show that for each fixed positive integer $k$, there are only finitely many $k$-crossing-critical simple graphs of average degree at least six. Combined with the recent constructions of crossing-critical graphs given by Bokal, this settles the question of for which numbers $q>0$ there is an infinite family of $k$-crossing-critical simple graphs of average degree $q$

Crossing-critical graphs with large maximum degree

A conjecture of Richter and Salazar about graphs that are critical for a fixed crossing number $k$ is that they have bounded bandwidth. A weaker well-known conjecture of Richter is that their maximum degree is bounded in terms of $k$. In this note we disprove these conjectures for every $k\ge 171$, by providing examples of $k$-crossing-critical graphs with arbitrarily large maximum degree

Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4

We find a graph of genus $5$ and its drawing on the orientable surface of genus $4$ with every pair of independent edges crossing an even number of times. This shows that the strong Hanani-Tutte theorem cannot be extended to the orientable surface of genus $4$. As a base step in the construction we use a counterexample to an extension of the unified Hanani-Tutte theorem on the torus.Comment: 12 pages, 4 figures; minor revision, new section on open problem