GD 2012: Graph Drawing pp 315-326

# 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

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## 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