Colorful Polytopes and Graphs

TL;DR

This paper explores the relationship between abstract polytopes and edge-colored graphs, introducing colorful polytopes that encode graph properties and examining their symmetries and applications in topology.

Contribution

It defines a new class of polytopes associated with edge-colored graphs and studies their properties, especially symmetry, linking combinatorics, geometry, and algebra.

Findings

Construction of colorful polytopes from edge-colored graphs

Analysis of symmetry properties of these polytopes

Applications to topological studies of manifolds

Abstract

The paper investigates connections between abstract polytopes and properly edge colored graphs. Given any finite n-edge-colored n-regular graph G, we associate to G a simple abstract polytope P_G of rank n, called the colorful polytope of G, with 1-skeleton isomorphic to G. We investigate the interplay between the geometric, combinatorial, or algebraic properties of the polytope P_G and the combinatorial or algebraic structure of the underlying graph G, focussing in particular on aspects of symmetry. Several such families of colorful polytopes are studied including examples derived from a Cayley graph, in particular the graphicahedra, as well as the flag adjacency polytopes and related monodromy polytopes associated with a given abstract polytope. The duals of certain families of colorful polytopes have been important in the topological study of colored triangulations and crystallization of manifolds.