On tangencies among planar curves with an application to coloring L-shapes

TL;DR

This paper establishes a linear bound on the number of tangencies among certain planar curves and applies this to improve the color bound for conflict-free coloring of grounded L-shapes.

Contribution

It proves a linear upper bound on tangencies among red and blue curves with limited intersections and improves the color bound for conflict-free coloring of grounded L-shapes from O(log^3 n) to O(log^2 n).

Findings

Linear bound on tangencies among specific planar curves.

Improved upper bound for conflict-free coloring of grounded L-shapes.

Application of geometric tangency bounds to coloring problems.

Abstract

We prove that there are $O(n)$ tangencies among any set of $n$ red and blue planar curves in which every pair of curves intersects at most once and no two curves of the same color intersect. If every pair of curves may intersect more than once, then it is known that the number of tangencies could be super-linear. However, we show that a linear upper bound still holds if we replace tangencies by pairwise disjoint connecting curves that all intersect a certain face of the arrangement of red and blue curves. The latter result has an application for the following problem studied by Keller, Rok and Smorodinsky [Disc.\ Comput.\ Geom.\ (2020)] in the context of \emph{conflict-free coloring} of \emph{string graphs}: what is the minimum number of colors that is always sufficient to color the members of any family of $n$ \emph{grounded L-shapes} such that among the L-shapes intersected by any L-shape there is one with a unique color? They showed that $O(\log^3 n)$ colors are always sufficient and that $\Omega(\log n)$ colors are sometimes necessary. We improve their upper bound to $O(\log^2 n)$.