Some properties of Bowlin and Brin's color graphs

TL;DR

This paper investigates the properties of color graphs derived from triangulated polygons with fixed four-colorings, proving their embeddability in hypercubes, providing bounds, and analyzing specific cases up to octagons.

Contribution

It establishes that all such color graphs can be embedded in hypercubes, provides an upper bound for the embedding dimension, and explores their structural features.

Findings

Each color graph has a vertex-induced hypercube embedding.

An upper bound for the hypercube dimension is provided.

Some color graphs cannot be embedded in any hypercube dimension.

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 some of their features are discussed. Finally it is shown that certain color graphs cannot be isometrically embedded in a hypercube of any dimension.