Grid representations and the chromatic number

Abstract

A grid drawing of a graph maps vertices to grid points and edges to line segments that avoid grid points representing other vertices. We show that there is a number of grid points that some line segment of an arbitrary grid drawing must intersect. This number is closely connected to the chromatic number. Second, we study how many columns we need to draw a graph in the grid, 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

Compact Grid Representation of Graphs

“…This problem has just been solved in the affirmative independently by M. Balko [1] and Flores-Peñaloza et al [8], although the grid size of the resulting representation is still excessively large. So it makes sense to find bounds on the size of the grid for some specific families of graphs, as we do below for bipartite graphs.…”

Grid Drawings and the 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 316 M. Balko…”