Grid Representations and the Chromatic Number

TL;DR

This paper explores the relationship between grid representations of graphs, the chromatic number, and introduces new NP-complete problems related to grid column requirements, also proving a conjecture about planar graph drawings.

Contribution

It establishes a connection between grid line intersections and the chromatic number, introduces NP-complete problems for grid column minimization, and proves a conjecture on planar grid drawings.

Findings

Some line segment in any grid drawing must intersect a number of grid points related to the chromatic number

New NP-complete problems for determining grid column requirements

Every planar graph has a planar grid drawing with segments containing exactly two grid points

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 David Flores-Pe\~naloza and Francisco Javier Zaragoza Martinez.