Some properties of Bowlin and Brin’s color graphs

Abstract

Bowlin and Brin defined the class of color graphs, whose vertices are triangulated polygons compatible with a fixed four-coloring of the polygon vertices. In this article it is proven that each color graph has a vertex-induced embedding in a hypercube, and an upper bound is given for the hypercube dimension. The color graphs for n-gons up to n=8 are listed and studied, in particular enabling a question by Bowlin and Brin concerning the diameter of color graphs to be answered. Finally it is shown that color graphs with a certain type of subgraph cannot be isometrically embedded in a hypercube of any dimension.