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# Long Alternating Paths Exist

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LIPIcs.SoCG.2020.57.pdf
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## Acknowledgements

This work was initiated at the second DACH workshop on Arrangements and Drawings which took place 21. - 25. January 2019 at Schloss St. Martin, Graz, Austria. We would like to thank the organizers and all the participants of the workshop for creating a conducive research atmosphere and for stimulating discussions. Part of this work was done on the Seventh Annual Workshop on Geometry and Graphs, Bellairs Research Institute, Holetown, Barbados, 10. - 15. March 2019. We also thank Zoltán Király for pointing out the reference [Clemens Müllner and Andrew Ryzhikov, 2019] to us.

## Cite As

Wolfgang Mulzer and Pavel Valtr. Long Alternating Paths Exist. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 57:1-57:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.57

## Abstract

Let P be a set of 2n points in convex position, such that n points are colored red and n points are colored blue. A non-crossing alternating path on P of length 𝓁 is a sequence p₁, … , p_𝓁 of 𝓁 points from P so that (i) all points are pairwise distinct; (ii) any two consecutive points p_i, p_{i+1} have different colors; and (iii) any two segments p_i p_{i+1} and p_j p_{j+1} have disjoint relative interiors, for i ≠ j. We show that there is an absolute constant ε > 0, independent of n and of the coloring, such that P always admits a non-crossing alternating path of length at least (1 + ε)n. The result is obtained through a slightly stronger statement: there always exists a non-crossing bichromatic separated matching on at least (1 + ε)n points of P. This is a properly colored matching whose segments are pairwise disjoint and intersected by common line. For both versions, this is the first improvement of the easily obtained lower bound of n by an additive term linear in n. The best known published upper bounds are asymptotically of order 4n/3+o(n).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• Non-crossing path
• bichromatic point sets

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## References

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