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On tangencies among planar curves with an application to coloring L-shapes
Combinatorics · Submitted 4 Mar 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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 $O(\log^2 n)$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | result 0.89 | 0.96 | 0.77 | asserted 1.00 |
| s1 | If every pair of curves may intersect more than once, then it is known that the number of tangencies could be super-linear. | background 0.97 | 0.06 | 0.03 | hedged possibility 0.70 |
| s2 | 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. | result 0.76 | 0.95 | 0.18 | asserted 1.00 |
| s3 | 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? | interpretation 0.45 | 0.24 | 0.08 | asserted 0.94 |
| s4 | They showed that $O(\log^3 n)$ colors are always sufficient and that $\Omega(\log n)$ colors are sometimes necessary. | background 0.99 | 0.03 | 0.04 | asserted 1.00 |
| s5 | We improve their upper bound to $O(\log^2 n)$. | result 0.78 | 0.96 | 0.32 | asserted 1.00 |
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Original 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 $O(\log^2 n)$.
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