Colorful Associahedra and Cyclohedra

TL;DR

This paper introduces colorful versions of associahedra and cyclohedra, which incorporate edge coloring in triangulations, providing a new combinatorial framework for understanding colored flips and triangulations.

Contribution

It defines colorful associahedra and cyclohedra as polytopes derived from edge-colored graphs, extending classical polytopes with color-preserving flip operations.

Findings

Colorful associahedra encode colored triangulations.

Colorful cyclohedra represent colored flip graphs.

The polytopes generalize classical associahedra and cyclohedra with coloring features.

Abstract

Every n-edge colored n-regular graph G naturally gives rise to a simple abstract n-polytope, the colorful polytope of G, whose 1-skeleton is isomorphic to G. The paper describes colorful polytope versions of the associahedron and cyclohedron. Like their classical counterparts, the colorful associahedron and cyclohedron encode triangulations and flips, but now with the added feature that the diagonals of the triangulations are colored and adjacency of triangulations requires color preserving flips. The colorful associahedron and cyclohedron are derived as colorful polytopes from the edge colored graph whose vertices represent these triangulations and whose colors on edges represent the colors of flipped diagonals.