Grid Drawings and the Chromatic Number

  • Martin Balko
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)

Abstract

A grid drawing of a graph maps vertices to the grid ℤd and edges to line segments that avoid grid points representing other vertices. We show that a graph G is qd-colorable, d, q ≥ 2, if and only if there is a grid drawing of G in ℤd in which no line segment intersects more than q grid points. This strengthens the result of D. Flores Pen̋aloza and F. J. Zaragoza Martinez. Second, we study grid drawings with a bounded number of columns, introducing some new NP-complete problems. Finally, we show that any planar graph has a planar grid drawing where every line segment contains exactly two grid points. This result proves conjectures asked by D. Flores Pen̋aloza and F. J. Zaragoza Martinez.

Keywords

graph drawings grid graph coloring chromatic number 

References

  1. 1.
    Balko, M.: Grid representations and the chromatic number. In: 28th European Workshop on Computational Geometry, EuroCG 2012, pp. 45–48 (2012)Google Scholar
  2. 2.
    Cáceres, J., Cortés, C., Grima, C.I., Hachimori, M., Márquez, A., Mukae, R., Nakamoto, A., Negami, S., Robles, R., Valenzuela, J.: Compact Grid Representation of Graphs. In: Márquez, A., Ramos, P., Urrutia, J. (eds.) EGC 2011. LNCS, vol. 7579, pp. 166–174. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Chappell, G.G., Gimbel, J., Hartman, C.: Thresholds for path colorings of planar graphs. Topics in Discrete Mathematics, 435–454 (2006)Google Scholar
  4. 4.
    Chrobak, M., Nakano, S.: Minimum-width grid drawings of plane graphs. Computational Geometry 11, 29–54 (1995)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Flores Pen̋aloza, D., Zaragoza Martinez, F.J.: Every four-colorable graph is isomorphic to a subgraph of the visibility graph of the integer lattice. In: Proceedings of the 21st Canadian Conference on Computational Geometry, CCCG 2009, pp. 91–94 (2009)Google Scholar
  6. 6.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Goddard, W.: Acyclic colorings of planar graphs. Discrete Math. 91, 91–94 (1991)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Kára, J., Pór, A., Wood, D.R.: On the chromatic number of the visibility graph of a set of points in the plane. Discrete Comput. Geom. 34, 497–506 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Lovász, L.: On decomposition of graphs. Studia Math. Hung. 1, 237–238 (1966)MATHGoogle Scholar
  10. 10.
    Pór, A., Wood, D.R.: No-three-in-line-in-3d. Algorithmica 47, 481–488 (2007)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1990, pp. 138–148. Society for Industrial and Applied Mathematics (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Martin Balko
    • 1
  1. 1.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPraha 1Czech Republic

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