Beyond the Richter-Thomassen Conjecture
نویسندگان
چکیده
If two closed Jordan curves in the plane have precisely one point in common, then it is called a touching point. All other intersection points are called crossing points. The main result of this paper is a Crossing Lemma for closed curves: In any family of n pairwise intersecting simple closed curves in the plane, no three of which pass through the same point, the number of crossing points exceeds the number of touching points by a factor of at least Ω((log logn)). As a corollary, we prove the following long-standing conjecture of Richter and Thomassen: The total number of intersection points between any n pairwise intersecting simple closed curves in the plane, no three of which pass through the same point, is at least (1− o(1))n. EPFL, Lausanne and Rényi Institute, Budapest. Supported by Swiss National Science Foundation Grants 200020144531 and 20021-137574. Email: [email protected] Ben Gurion University of the Negev, Beer-Sheba, Israel. Email: [email protected]. Supported by Minerva Fellowship Program of the Max Planck Society, by the Fondation Sciences Mathématiques de Paris (FSMP), and by a public grant overseen by the French National Research Agency (ANR) as part of the Investissements dAvenir program (reference: ANR-10-LABX-0098). Rényi Institute, Budapest. Supported by the “Lendület” Project of the Hungarian Academy of Sciences. Email: [email protected]
منابع مشابه
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. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the f...
متن کاملComputing 2-Walks in Polynomial Time
A 2-walk of a graph is a walk visiting every vertex at least once and at most twice. By generalizing decompositions of Tutte and Thomassen, Gao, Richter and Yu proved that every 3-connected planar graph contains a closed 2-walk such that all vertices visited twice are contained in 3separators. This seminal result generalizes Tutte’s theorem that every 4-connected planar graph is Hamiltonian as ...
متن کاملEdge-partitioning a graph into paths: beyond the Barát-Thomassen conjecture
The Barát-Thomassen conjecture asserts that there is a function f such that for every fixed tree T with t edges, every graph which is f(t)edge-connected with its number of edges divisible by t has a partition of its edges into copies of T . This has been proved in the case of paths of length 2 by Thomassen [14], and recently shown to be true for all paths by Botler, Mota, Oshiro and Wakabayashi...
متن کاملA proof of the Barát-Thomassen conjecture
The Barát-Thomassen conjecture asserts that for every tree T on m edges, there exists a constant kT such that every kT -edge-connected graph with size divisible by m can be edge-decomposed into copies of T . So far this conjecture has only been verified when T is a path or when T has diameter at most 4. Here we prove the full statement of the conjecture.
متن کاملDecomposing Highly Connected Graphs into Paths of Length Five
Barát and Thomassen (2006) posed the following decomposition conjecture: for each tree T , there exists a natural number kT such that, if G is a kT -edge-connected graph and |E(G)| is divisible by |E(T )|, then G admits a decomposition into copies of T . In a series of papers, Thomassen verified this conjecture for stars, some bistars, paths of length 3, and paths whose length is a power of 2. ...
متن کامل