On the number of touching pairs in a set of planar curves

Györgyi, Péter; Hujter, Bálint; Kisfaludi-Bak, Sándor

doi:10.1016/j.comgeo.2017.10.004

Cited by 7 publications

(4 citation statements)

References 16 publications

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“…According to a conjecture of Pach [10] the number of tangencies between an at most 1-intersecting family of n curves is O(n) if every pair of curves intersects. The conjecture is known to hold for pseudo-circles [3], and in the special case when there exist constantly many faces of the arrangement of the curves such that every curve has one of its endpoints inside one of these faces [8].…”

Section: Curvementioning

confidence: 99%

The number of tangencies between two families of curves

Keszegh¹,

Pálvölgyi²

2021

Preprint

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Section: Curvementioning

confidence: 99%

The number of tangencies between two families of curves

Keszegh¹,

Pálvölgyi²

2021

Preprint

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“…Each intersection point involves exactly two curves and is either a proper crossing of these curves or an endpoint of one of them that belongs to the interior of the other curve. 7 2 Proof of Theorem 1…”

Section: Thus We Havementioning

confidence: 99%

“…1 According to a nice conjecture of Pach [13] the number of tangencies among a 1-intersecting family S of n curves should be O(n) if every pair of curves intersects. Györgyi et al [7] proved this conjecture in the special case where there are constantly many faces of the arrangement of S such that every curve in S has one of its endpoints inside one of these faces. Here we prove the following variant.…”

Section: Introductionmentioning

confidence: 98%

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

Ackerman¹,

Keszegh²,

Pálvölgyi³

2021

Preprint

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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 Lshapes 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 Ω(log n) colors are sometimes necessary. We improve their upper bound to O(log 2 n).

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“…Györgyi, Hujter and Kisfaludi-Bak [7] proved Conjecture 1 for the special case where there are constantly many faces in the arrangement of C that together contain all the endpoints of the curves. In this paper we show that Conjecture 1 also holds for x-monotone curves.…”

Section: Introductionmentioning

confidence: 99%