Quadrangulations and 2-Colorations

Abstract

Any metric quadrangulation (made by segments of straight line) of a point set in the plane determines a 2-coloration of the set, such that edges of the quadrangulation can only join points with different colors. In this work we focus in 2-colorations and study whether they admit a quadrangulation or not, and whether, given two quadrangulations of the same 2-coloration, it is possible to carry one into the other using some local operations, called diagonal slides and diagonal rotation. Although the answer is negative in general, we can show a very wide family of 2-colorations, called onions 2-coloration, that are quadrangulable and which graph of quadrangulations is always connected.