Grid representations and the chromatic number

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.