## 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 Penaloza 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 Penaloza and F.J. Zaragoza Martinez.

Original language | English |
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Pages (from-to) | 990-1002 |

Number of pages | 13 |

Journal | Computational Geometry: Theory and Applications |

Volume | 46 |

Issue number | 8 |

DOIs | |

State | Published - 22 May 2013 |

Externally published | Yes |

## Keywords

- Chromatic number
- Graph coloring
- Graph drawings
- Grid

## ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics