Abstract
Given a line arrangement \(\mathcal A\) with n lines, we show that there exists a path of length \(n^2/3 - O(n)\) in the dual graph of \(\mathcal A\) formed by its faces. This bound is tight up to lower order terms. For the bicolored version, we describe an example of a line arrangement with 3k blue and 2k red lines with no alternating path longer than 14k. Further, we show that any line arrangement with n lines has a coloring such that it has an alternating path of length \(\varOmega (n^2/ \log n)\). Our results also hold for pseudoline arrangements.
U. Hoffmann—Supported by the Deutsche Forschungsgemeinschaft within the research training group ‘Methods for Discrete Structures’ (GRK 1408).
T. Miltzow—Supported by the ERC grant PARAMTIGHT: Parameterized complexity and the search for tight complexity results\(''\), no. 280152.
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References
Aichholzer, O., Cardinal, J., Hackl, T., Hurtado, F., Korman, M., Pilz, A., Silveira, R., Uehara, R., Valtr, P., Vogtenhuber, B., Welzl, E.: Cell-paths in mono-and bichromatic line arrangements in the plane. Discrete Math. Theor. Comput. Sci. 16(3), 317–332 (2014)
Felsner, S.: Geometric Graphs and Arrangements. Vieweg, Verlag (2004)
Füredi, Z., Palásti, I.: Arrangements of lines with a large number of triangles. Proc. Am. Math. Soc. 92(4), 561–566 (1984)
Goodman, J.E., O’Rourke, J. (eds.): Handbook of Discrete and Computational Geometry. CRC Press, Boca Raton (2010)
Hoffmann, U., Kleist, L., Miltzow, T.: Long paths in line arrangements. Eur. Workshop Comput. Geom. 2015, 157–160 (2015)
Hoffmann, U., Kleist, L., Miltzow, T.: Upper and lower bounds on long dual-paths in line arrangements. ArXiv e-prints 1506.03728 (2015)
Tutte, W.T.: A theorem on planar graphs. Trans. Am. Math. Soc. 82, 99–116 (1956)
Acknowledgment
We thank Nieke Aerts, Stefan Felsner, Heuna Kim, and Piotr Micek for interesting and helpful discussions on the topic. We thank the anonymous reviewers for helpful comments.
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Hoffmann, U., Kleist, L., Miltzow, T. (2015). Upper and Lower Bounds on Long Dual Paths in Line Arrangements. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_34
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DOI: https://doi.org/10.1007/978-3-662-48054-0_34
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