Abstract
We study some variants of Conway’s thrackle conjecture. A tangle is a graph drawn in the plane such that its edges are represented by continuous arcs, and any two edges share precisely one point, which is either a common endpoint or an interior point at which the two edges are tangent to each other. These points of tangencies are assumed to be distinct. If we drop the last assumption, that is, more than two edges may touch one another at the same point, then the drawing is called a degenerate tangle. We settle a problem of Pach, Radoičić, and Tóth [7], by showing that every degenerate tangle has at most as many edges as vertices. We also give a complete characterization of tangles.
Chapter PDF
Similar content being viewed by others
References
Unsolved problems. Chairman: P. Erdős. In: Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), Inst. Math. Appl., Southend-on-Sea, pp. 351–363 (1972)
Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)
Cairns, G., McIntyre, M., Nikolayevsky, Y.: The thrackle conjecture for \(K\sb 5\) and \(K\sb{3,3}\). In: Towards a Theory of Geometric Graphs, Contemp. Math., vol. 342, pp. 35–54. Amer. Math. Soc., Providence (2004)
Fulek, R., Pach, J.: A computational approach to Conway’s thrackle conjecture. Comput. Geom. 44(6-7), 345–355 (2011)
Kuratowski, K.: Sur le probleme des courbes gauches en topologie. Fund. Math. 15, 271–283 (1930)
Lovász, L., Pach, J., Szegedy, M.: On Conway’s thrackle conjecture. Discrete Comput. Geom. 18, 369–376 (1998)
Pach, J., Radoičić, R., Tóth, G.: Tangled thrackles. Geombinatorics (2012) (to appear)
Pach, J., Tóth, G.: Disjoint Edges in Topological Graphs. In: Akiyama, J., Baskoro, E.T., Kano, M. (eds.) IJCCGGT 2003. LNCS, vol. 3330, pp. 133–140. Springer, Heidelberg (2005)
Ringeisen, R.D.: Two old extremal graph drawing conjectures: progress and perspectives. Congressus Numerantium 115, 91–103 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ruiz-Vargas, A.J. (2013). Tangles and Degenerate Tangles. In: Didimo, W., Patrignani, M. (eds) Graph Drawing. GD 2012. Lecture Notes in Computer Science, vol 7704. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36763-2_31
Download citation
DOI: https://doi.org/10.1007/978-3-642-36763-2_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36762-5
Online ISBN: 978-3-642-36763-2
eBook Packages: Computer ScienceComputer Science (R0)