International Symposium on Mathematical Foundations of Computer Science

MFCS 2015: Mathematical Foundations of Computer Science 2015 pp 407-419 | Cite as

Upper and Lower Bounds on Long Dual Paths in Line Arrangements

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9235)


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.


  1. 1.
    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)MathSciNetMATHGoogle Scholar
  2. 2.
    Felsner, S.: Geometric Graphs and Arrangements. Vieweg, Verlag (2004)CrossRefMATHGoogle Scholar
  3. 3.
    Füredi, Z., Palásti, I.: Arrangements of lines with a large number of triangles. Proc. Am. Math. Soc. 92(4), 561–566 (1984)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Goodman, J.E., O’Rourke, J. (eds.): Handbook of Discrete and Computational Geometry. CRC Press, Boca Raton (2010)Google Scholar
  5. 5.
    Hoffmann, U., Kleist, L., Miltzow, T.: Long paths in line arrangements. Eur. Workshop Comput. Geom. 2015, 157–160 (2015)MATHGoogle Scholar
  6. 6.
    Hoffmann, U., Kleist, L., Miltzow, T.: Upper and lower bounds on long dual-paths in line arrangements. ArXiv e-prints 1506.03728 (2015)Google Scholar
  7. 7.
    Tutte, W.T.: A theorem on planar graphs. Trans. Am. Math. Soc. 82, 99–116 (1956)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Technische Universität BerlinBerlinGermany
  2. 2.Freie Universität BerlinBerlinGermany

Personalised recommendations