Abstract
A tangle is a graph drawn in the plane so that any pair of edges have precisely one point in common, and this point is either an endpoint or a point of tangency. If we allow a third option: the common point may be a proper crossing between the two edges, then the graph is called a tangled thrackle. We establish the following analogues of Conway’s thrackle conjecture: The number of edges of a tangle cannot exceed its number of vertices, n. We also prove that the number of edges of an x-monotone tangled thrackle with n vertices is at most n + 1. Both results are tight for n > 3. For not necessarily x-monotone tangled thrackles, we have a somewhat weaker, but nearly linear, upper bound.
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References
Ackerman, E., Fox, J., Pach, J., Suk, A.: On grids in topological graphs. In: Proceedings of the 25th Annual Symposium on Computational Geometry, pp. 403–412. ACM Press (2009)
Braß, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)
Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete and Computational Geometry 23, 191–206 (2000)
Cairns, G., McIntyre, M., Nikolayevsky, Y.: The thrackle conjecture for K5 and K3,3. In: Towards a Theory of Geometric Graphs, Contemp. Math., vol. 342, pp. 35–54. Amer. Math. Soc., Providence (2004)
Cairns, G., Nikolayevsky, Y.: Generalized thrackle drawings of non-bipartite graphs. Discrete and Computational Geometry 41, 119–134 (2009)
Cairns, G., Nikolayevsky, Y.: Outerplanar thrackles. Graphs and Combinatorics 28, 85–96 (2012)
Fox, J., Frati, F., Pach, J., Pinchasi, R.: Crossings between Curves with Many Tangencies. In: Rahman, M. S., Fujita, S. (eds.) WALCOM 2010. LNCS, vol. 5942, pp. 1–8. Springer, Heidelberg (2010)
Fulek, R., Pach, J.: A computational approach to Conway’s thrackle conjecture. Computational Geometry: Theory and Applications 44, 345–355 (2011)
Graham, R.L.: The largest small hexagon. Journal of Combinatorial Theory, Series A 18, 165–170 (1975)
Green, J.E., Ringeisen, R.D.: Combinatorial drawings and thrackle surfaces. In: Graph Theory, Combinatorics, and Algorithms (Kalamazoo, MI, 1992), vol. 2, pp. 999–1009. Wiley-Intersci. Publ., Wiley, New York (1995)
Hopf, H., Pannwitz, E.: Aufgabe Nr. 167. Jahresbericht Deutsch. Math.-Verein. 43, 114 (1934)
Lovász, L., Pach, J., Szegedy, M.: On Conway’s thrackle conjecture. Discrete and Computational Geometry 18, 369–376 (1998)
Perlstein, A., Pinchasi, R.: Generalized thrackles and geometric graphs in ℝ3 with no pair of strongly avoiding edges. Graphs and Combinatorics 24, 373–389 (2008)
Piazza, B.L., Ringeisen, R.D., Stueckle, S.K.: Subthrackleable graphs and four cycles. In: Graph Theory and Applications (Hakone, 1990). Discrete Mathematics, vol. 127, pp. 265–276 (1994)
Pach, J., Sterling, E.: Conway’s conjecture for monotone thrackles. American Mathematical Monthly 118, 544–548 (2011)
Pach, J., Suk, A., Treml, M.: Tangencies between families of disjoint regions in the plane. Computational Geometry: Theory and Applications 45, 131–138 (2012)
Pach, J., Törőcsik, J.: Some geometric applications of Dilworth’s theorem. Discrete and Computational Geometry 12, 1–7 (1994)
Pach, J., Tóth, G.: Disjoint edges in topological graphs. Journal of Combinatorics 1, 335–344 (2010)
Ringeisen, R.D.: Two old extremal graph drawing conjectures: progress and perspectives. Congressus Numerantium 115, 91–103 (1996)
Tóth, G.: Note on geometric graphs. Journal of Combinatorial Theory, Series A 89, 126–132 (2000)
Erdős, P.: Unsolved problem. In: Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), pp. 351–363. Inst. Math. Appl., Southend-on-Sea (1972)
Woodall, D.R.: Thrackles and deadlock. In: Welsh, D.J.A. (ed.) Combinatorial Mathematics and Its Applications, pp. 335–347. Academic Press (1969)
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Pach, J., Radoičić, R., Tóth, G. (2012). Tangled Thrackles. In: Márquez, A., Ramos, P., Urrutia, J. (eds) Computational Geometry. EGC 2011. Lecture Notes in Computer Science, vol 7579. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34191-5_4
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DOI: https://doi.org/10.1007/978-3-642-34191-5_4
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