On Tangencies Among Planar Curves with an Application to Coloring L-Shapes

Ackerman, Eyal; Keszegh, Balázs; Pálvölgyi, Dömötör

doi:10.1007/978-3-030-83823-2_20

Cited by 6 publications

(5 citation statements)

References 25 publications

(19 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We need one more technical lemma about hypergraphs with the HLD property. It was explicitly stated in [2], based on earlier ideas from [4, Lem. 2.6] and [5], and can be viewed as an abstract version of the Clarkson-Shor technique [9]: E) be a hypergraph with the HLD property for some parameter c ∈ R >0 .…”

Section: Linear Delaunay Graphsmentioning

confidence: 99%

Conflict-Free Colouring of Subsets

Jartoux¹,

Keller²,

Smorodinsky³

et al. 2022

Preprint

View full text Add to dashboard Cite

We introduce and study conflict-free colourings of t-subsets in hypergraphs. In such colourings, one assigns colours to all subsets of vertices of cardinality t such that in any hyperedge of cardinality at least t there is a uniquely coloured t-subset.The case t = 1, i.e., vertex conflict-free colouring, is a well-studied notion that has been originated in the context of frequency allocation to antennas in cellular networks. It has caught the attention of researchers both from the combinatorial and algorithmic points of view. A special focus was given to hypergraphs arising in geometry.Already the case t = 2 (i.e., colouring pairs) seems to present a new challenge. Many of the tools used for conflict-free colouring of geometric hypergraphs rely on hereditary properties of the underlying hypergraphs. When dealing with subsets of vertices, the properties do not pass to subfamilies of subsets. Therefore, we develop new tools, which might be of independent interest.(i) For any fixed t, we show that the n t t-subsets in any set P of n points in the plane can be coloured with O(t 2 log 2 n) colours so that any axis-parallel rectangle that contains at least t points of P also contains a uniquely coloured t-subset.(ii) For a wide class of "well behaved" geometrically defined hypergraphs, we provide near tight upper bounds on their t-subset conflict-free chromatic number.For t = 2 we show that for each of those "well -behaved" hypergraphs H, the hypergraph H obtained by taking union of two hyperedges from H, admits a 2-subset conflict-free colouring with roughly the same number of colours as H. For example, we show that the n 2 pairs of points in any set P of n points in the plane can be coloured with O(log n) colours such that for any two discs d1, d2 in the plane with |(d1 ∪ d2) ∩ P | ≥ 2 there is a uniquely (in d1 ∪ d2) coloured pair.(iii) We also show that there is no general bound on the t-subset conflict-free chromatic number as a function of the standard conflict-free chromatic number already for t = 2.

show abstract

Section: Linear Delaunay Graphsmentioning

confidence: 99%

Conflict-Free Colouring of Subsets

Jartoux¹,

Keller²,

Smorodinsky³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…The number of tangencies within an at most 1-intersecting family of n curves can be Ω(n 4/3 ), even if all curves are required to be x-monotone; this follows from a famous construction of Erdős and Purdy [5] of n points and n lines that determine Ω(n 4/3 ) point-line incidences by replacing each point with a small curve, and slightly pertubing the lines; see [11]. For x-monotone curves it is mentioned in [2] that an almost matching upper bound of O(n 4/3 log 2/3 n) follows from a result of Pach and Sharir [14] which can be improved with a more careful analysis to O(n 4/3 log 1/3 n). For further results, see [4,6,19].…”

Section: Curvementioning

confidence: 99%

“…This raised the question if assuming that the family is at most 1-intersecting leads to a better upper bound. Ackerman et nos [2] proved that this is indeed the case, i.e., given an at most 1-intersecting family of n red and blue curves such that no two curves of the same color intersect, the number of tangencies between the curves is O(n). Note that we do not need to assume x-monotonicity and that it is trivial to construct an example with Ω(n) tangencies.…”

Section: Introductionmentioning

confidence: 96%

“…In case of a doubly-grounded family, this is trivial 2. In fact, it would be sufficient to assume that u 2 v 2 is an edge of G, as since there is no self-crossing C 4 , this can only happen if u 2 v 2 is an edge of the C 6 .…”

mentioning

confidence: 99%

“…u 3 are on the left side of the strip in this order, v 1 , v 2 , v 3 are on the right side in this order, and u 2 v 2 is an edge of the C 6 . 2 Assume on the contrary that such a positive C 6 exists. Recall that each edge u i v j has a red part incident to u i and a blue part incident to v j , and that no parts of the same color can cross.…”

mentioning

confidence: 99%

See 2 more Smart Citations