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Grid Drawings and the Chromatic Number

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Part of the Lecture Notes in Computer Science book series (LNTCS,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 q d-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

A four-page abstract of an earlier version of this paper appeared in the proceedings of the 28th European Workshop on Computational Geometry EuroCG ’12 [1]. The booklet of abstracts is available here: http://www.diei.unipg.it/eurocg2012/ booklet.pdf

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Balko, M. (2013). Grid Drawings and the Chromatic Number. In: Didimo, W., Patrignani, M. (eds) Graph Drawing. GD 2012. Lecture Notes in Computer Science, vol 7704. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36763-2_28

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  • DOI: https://doi.org/10.1007/978-3-642-36763-2_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36762-5

  • Online ISBN: 978-3-642-36763-2

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