# Many Touchings Force Many Crossings

• János Pach
• Géza Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10692)

## Abstract

Given n continuous open curves in the plane, we say that a pair is touching if they have only one interior point in common and at this point the first curve does not get from one side of the second curve to its other side. Otherwise, if the two curves intersect, they are said to form a crossing pair. Let t and c denote the number of touching pairs and crossing pairs, respectively. We prove that $$c \ge {1\over 10^5}{t^2\over n^2}$$, provided that $$t\ge 10n$$. Apart from the values of the constants, this result is best possible.

## Keywords

Planar curves Touching Crossing

## Notes

### Acknowledgment

The work of János Pach was partially supported by Swiss National Science Foundation Grants 200021-165977 and 200020-162884. Géza Tóth’s work was partially supported by the Hungarian National Research, Development and Innovation Office, NKFIH, Grant K-111827.

## References

1. 1.
Ajtai, M., Chvátal, V., Newborn, M., Szemerédi, E.: Crossing-free subgraphs. In: Theory and Practice of Combinatorics. North-Holland Mathematics Studies, vol. 60, pp. 9–12, North-Holland, Amsterdam (1982)Google Scholar
2. 2.
Dey, T.K.: Improved bounds on planar $$k$$-sets and related problems. Discrete Comput. Geom. 19, 373–382 (1998)
3. 3.
Ellenberg, J. S., Solymosi, J., Zahl, J.: New bounds on curve tangencies and orthogonalities. Discrete Anal., Paper No. 18, 22 pp. (2016)Google Scholar
4. 4.
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).
5. 5.
Fox, J., Pach, J.: A separator theorem for string graphs and its applications. Comb. Probab. Comput. 19, 371–390 (2010)
6. 6.
García, A., Noy, M., Tejel, J.: Lower bounds on the number of crossing-free subgraphs of $$K_n$$. Comput. Geom. 16(4), 211–221 (2000)
7. 7.
Leighton, T.: Complexity Issues in VLSI, Foundations of Computing Series. MIT Press, Cambridge (1983)Google Scholar
8. 8.
Pach, J., Rubin, N., Tardos, G.: Beyond the Richter-Thomassen Conjecture. In: Proceedings of 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, pp. 957–968. SIAM (2016)Google Scholar
9. 9.
Pach, J., Tóth, G.: How many ways can one draw a graph? Combinatorica 26(5), 559–576 (2006)
10. 10.
Sharir, M.: The Clarkson-Shor technique revisited and extended. Comb. Prob. Comput. 12(2), 191–201 (2003)
11. 11.
Solymosi, J., Tóth, C.D.: Distinct distances in the plane. Discrete Comput. Geom. 25(4), 629–634 (2001)
12. 12.
Székely, L.A.: Crossing numbers and hard Erdős problems in discrete geometry. Comb. Prob. Comput. 6(3), 353–358 (1997)
13. 13.
Tutte, W.T.: Toward a theory of crossing numbers. J. Comb. Theory 8, 45–53 (1970)