Non-homotopic Loops with a Bounded Number of Pairwise Intersections

0551784 - ÚI 2022 RIV CH eng C - Konferenční příspěvek (zahraniční konf.)
Blažej, V. - Opler, M. - Šileikis, Matas - Valtr, P.
Non-homotopic Loops with a Bounded Number of Pairwise Intersections.
Graph Drawing and Network Visualization. 29th International Symposium GD 2021, Revised Selected Papers. Cham: Springer, 2021 - (Purchase, H.; Rutter, I.), s. 210-222. Lecture Notes in Computer Science, 12868. ISBN 978-3-030-92930-5. ISSN 0302-9743.
[GD 2021: International Symposium on Graph Drawing and Network Visualization /29./. Tübingen (DE), 14.09.2021-17.09.2021]
Grant CEP: GA ČR(CZ) GJ20-27757Y
Institucionální podpora: RVO:67985807
Klíčová slova: Graph drawing * Non-homotopic loops * Curve intersections * Plane
Obor OECD: Pure mathematics

Let V_n be a set of n points in the plane and let x∈V_n . An x-loop is a continuous closed curve not containing any point of V_n . We say that two x-loops are non-homotopic if they cannot be transformed continuously into each other without passing through a point of Vn . For n=2, we give an upper bound e^O(k^(1/2)) on the maximum size of a family of pairwise non-homotopic x-loops such that every loop has fewer than k self-intersections and any two loops have fewer than k intersections. The exponent O(k^(1/2)) is asymptotically tight. The previous upper bound 2^((2k)^4) was proved by Pach et al. [6]. We prove the above result by proving the asymptotic upper bound e^O(k^(1/2)) for a similar problem when x∈V_n, and by proving a close relation between the two problems.
Trvalý link: http://hdl.handle.net/11104/0327001