On tangencies among planar curves with an application to coloring L-shapes
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 sup...
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| Published in | European journal of combinatorics Vol. 121; p. 103837 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
01.10.2024
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| Online Access | Get full text |
| ISSN | 0195-6698 1095-9971 1095-9971 |
| DOI | 10.1016/j.ejc.2023.103837 |
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| 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 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 ngrounded 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(logn) colors are sometimes necessary. We improve their upper bound to O(log2n). |
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| AbstractList | 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 ngrounded 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(logn) colors are sometimes necessary. We improve their upper bound to O(log2n). |
| ArticleNumber | 103837 |
| Author | Ackerman, Eyal Keszegh, Balázs Pálvölgyi, Dömötör |
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| Cites_doi | 10.1016/j.comgeo.2020.101687 10.1137/0220029 10.1137/1.9781611973099.125 10.1007/s00454-005-1162-6 10.1016/j.jctb.2013.11.001 10.37236/8096 10.1016/0012-365X(95)00316-O 10.1016/j.comgeo.2017.10.004 10.1016/j.dam.2016.01.028 10.1016/j.comgeo.2011.10.002 10.1007/s00454-019-00097-8 |
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| Title | On tangencies among planar curves with an application to coloring L-shapes |
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