Topic
Thrackle
About: Thrackle is a research topic. Over the lifetime, 37 publications have been published within this topic receiving 589 citations.
Papers
1 Jan 2004
TL;DR: On the complexity of the linkage reconfiguration problem by H. Alt, C. Knauer, G. Rote, and S. Toth as mentioned in this paper, a new entropy inequality for the Erdos distance problem by N. Woyczynski.
Abstract: On the complexity of the linkage reconfiguration problem by H. Alt, C. Knauer, G. Rote, and S. Whitesides Falconer conjecture, spherical averages and discrete analogs by G. Arutyunyants and A. Iosevich Turan-type extremal problems for convex geometric hypergraphs by P. Brass The thrackle conjecture for $K_5$ and $K_{3,3}$ by G. Cairns, M. McIntyre, and Y. Nikolayevsky Three-dimensional grid drawings with sub-quadratic volume by V. Dujmovic and D. R. Wood On a coloring problem for the integer grid by A. Dumitrescu and R. Radoicic Separating thickness from geometric thickness by D. Eppstein Direction trees in centered polygons by R. E. Jamison Path coverings of two sets of points in the plane by A. Kaneko, M. Kano, and K. Suzuki Length of sums in a Minkowski space by G. O. H. Katona, R. Mayer, and W. A. Woyczynski A new entropy inequality for the Erdos distance problem by N. H. Katz and G. Tardos Coloring intersection graphs of geometric figures with a given clique number by A. Kostochka Convex quadrilaterals and $k$-sets by L. Lovasz, K. Vesztergombi, U. Wagner, and E. Welzl Distance graphs and rigidity by H. Maehara A Ramsey property of planar graphs by J. Nesetril, J. Solymosi, and P. Valtr A generalization of quasi-planarity by J. Pach, R. Radoicic, and G. Toth Geometric incidences by J. Pach and M. Sharir Large sets must have either a $k$-edge or a $(k+2)$-edge by M. A. Perles and R. Pinchasi Topological graphs with no self-intersecting cycle of length 4 by R. Pinchasi and R. Radoicic A problem on restricted sumsets by I. Z. Ruzsa The gap between crossing numbers and convex crossing numbers by F. Shahrokhi, O. Sykora, L. A. Szekely, and I. Vrto Distinct distances in high dimensional homogeneous sets by J. Solymosi and V. Vu The biplanar crossing number of the random graph by J. Spencer The unit distance problem on spheres by K. J. Swanepoel and P. Valtr Short proof for a theorem of Pach, Spencer, and Toth by L. Szekely.
268 citations
1 Sep 1995
TL;DR: It is shown that a thrackle has at most twice as many edges as vertices as it was conjectured that it cannot exceed the number of its vertices.
Abstract: A thrackle is a graph drawn in the plane so that its edges are represented by Jordan arcs and any two distinct arcs either meet at exactly one common vertex or cross at exactly one point interior to both arcs. About 40 years ago, J. H. Conway conjectured that the number of edges of a thrackle cannot exceed the number of its vertices. We show that a thrackle has at most twice as many edges as vertices. Some related problems and generalizations are also considered.
45 citations
TL;DR: It is shown that a non-bipartite graph can be drawn as a generalized thrackle on an oriented closed surface M if and only if there is a parity embedding of G in a closed non-orientable surface of Euler characteristic χ(M)−1.
Abstract: A graph drawing is called a generalized thrackle if every pair of edges meets an odd number of times. In a previous paper, we showed that a bipartite graph G can be drawn as a generalized thrackle on an oriented closed surface M if and only if G can be embedded in M. In this paper, we use Lins’ notion of a parity embedding and show that a non-bipartite graph can be drawn as a generalized thrackle on an oriented closed surface M if and only if there is a parity embedding of G in a closed non-orientable surface of Euler characteristic χ(M)−1. As a corollary, we prove a sharp upper bound for the number of edges of a simple generalized thrackle.
25 citations
TL;DR: A new, topological proof that the weak Hanani-Tutte theorem is true on orientable surfaces is given and a result of Pach and Toth is extended that every bipartite generalized thrackle in a surface S can be embedded in S.
16 citations
1 Jan 2018
TL;DR: This paper disproves the conjecture that any graph has a convex straight-line drawing, that is, a drawing with vertices in convex position, that maximizes the number of edge crossings and strengthens these results by showing hardness of approximation even for the unweighted geometric case and proving that the un Weighted topological case is NP-hard.
Abstract: Research about crossings is typically about minimization. In this paper, we consider maximizing the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a convex straight-line drawing, that is, a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case and prove that the unweighted topological case is NP-hard.
9 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 2 |
| 2019 | 3 |
| 2018 | 3 |
| 2017 | 4 |
| 2016 | 3 |
| 2015 | 1 |