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 conflict-free coloring of string graphs: what is the minimum number of colors that is always sufficient to color the members of any family of n 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(log3n) colors are always sufficient and that Ω(log n) colors are sometimes necessary. We improve their upper bound to O(log2n).
|Title of host publication||Trends in Mathematics|
|Publisher||Springer Science and Business Media Deutschland GmbH|
|Number of pages||6|
|State||Published - 2021|
|Name||Trends in Mathematics|
Bibliographical noteFunding Information:
Acknowledgement. We thank Chaya Keller for pointing out that Theorem 7 implies Theorem 8. Research of the second and third author supported by the Lendület program of the Hungarian Academy of Sciences, under the grant LP2017-19/2017. Research of the second author supported by the National Research, Development and Innovation Office – NKFIH under the grant K 132696.
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.
- Conflict-free coloring
- Geometric hypergraph
ASJC Scopus subject areas
- Mathematics (all)