Many Touchings Force Many Crossings

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.

7 pages; Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017)