On the maximum number of tangencies among $1$-intersecting curves

Abstract

According to a conjecture of Pach, there are $O(n)$ tangent pairs among any family of $n$ Jordan arcs in which every pair of arcs has precisely one common point and no three arcs share a common point. This conjecture was proved for two special cases, however, for the general case the currently best upper bound is only $O(n^{7/4})$. This is also the best known bound on the number of tangencies in the relaxed case where every pair of arcs has \emph{at most} one common point. We improve the bounds for the latter and former cases to $O(n^{5/3})$ and $O(n^{3/2})$, respectively. We also consider a few other variants of these questions, for example, we show that if the arcs are \emph{$x$-monotone}, each pair intersects at most once and their left endpoints lie on a common vertical line, then the maximum number of tangencies is $Θ(n^{4/3})$. Without this last condition the number of tangencies is $O(n^{4/3}(\log n)^{1/3})$, improving a previous bound of Pach and Sharir. Along the way we prove a graph-theoretic theorem which extends a result of Erdős and Simonovits and may be of independent interest.

Figures (9)

• Figure 1: The four tangency types for two oriented curves $c_1$ and $c_2$: (a) left-left (b) left-right (c) right-right (d) right-left.
• Figure 2: If $c_1$ and $c_2$ are grounded at $\ell$ and have the same tangency type with $c$, then $c_1[c,+]$ and $c_2[c,+]$ cannot cross.
• Figure 3: If the subcurves $a_i(b_1,b_2)$ were disjoint, then we could draw a crossing-free copy of $K_{2,3}$.
• Figure 4: Illustrations for the proof of Proposition \ref{['prop:abc-abc']}: $a_1[b_1,+]$ and $a_2[-,b_1]$ are crossing and together with $b_1[a_1,a_2]$ and $b_2[a_1,a_2]$ induce a partition of the plane. $a_1^-$ and $a_2^-$ lie on different regions of this partition for any red-blue tangency type.
• Figure 5: An illustration for the proof of Theorem \ref{['thm:pw-bi-infinite']}: If $G_{\cal L}$ contains a cycle, then consider the blue curve $b$ that starts below all the other blue curves in the cycle.

Theorems & Definitions (58)

• Conjecture 1: Pach pachpc
• Theorem 1: KP23
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7: ES70
• Theorem 8
• Theorem 9