Colorful polytopes and graphs

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, the colorful polytope of G, with 1-skeleton isomorphic to G. We investigate the interplay between the geometric, combinatorial, or algebraic properties of * garaujo@matem.unam.mx, 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.

On Roli's cube

“…We therefore say that P is a colourful polytope. Such polytopes were introduced in [1]. In general, one begins with a finite, connected dvalent graph G admitting a (proper) edge colouring, say by the symbols 1, .…”

On Roli's Cube

“…We therefore say that P is a colourful polytope. Such polytopes were introduced in [1]. In general, one begins with a finite, connected d-valent graph G admitting a (proper) edge colouring, say by the symbols 1, .…”

“…3,2,1);σ 2 = (1, 1, 1, 1) • (1, 2, 4); σ 3 = (1, 1, 1, 1) • (2, 4, 3),andσ 1 = (1, 1, 1, −1) • (1, 2, 3, 4); σ 2 = (−1, −1, 1, 1)(1, 3, 2); σ 3 = (1, 1, 1, 1) • (2, 4, 3).…”

A Finite Chiral 4-Polytope in $${\mathbb {R}}^4$$ R 4

“…As it is shown in Figure 1, we can colour the edges of K 4,4 with 4 colours in such a way that two edges of the same colour are not incident, so that each colour induces a perfect matching in the graph. We label the vertices of the It is not difficult to see that with the colouring of K 4,4 given in Figure 1, we can obtain a colourful 4-polytope P in the sense of [1]. In fact, the 2-faces of P are the cycles of K 4,4 that have exactly two colours.…”

“…(In fact, we observe that the graph Q 3 of the cube is precisely K 4,4 minus a perfect matching.) The automorphisms of P are all the colour respecting automorphisms of K 4,4 , that is, all the automorphisms of K 4,4 that induce a permutation on the colours (see [1]). Therefore, P is isomorphic to the hemi-hypercube {4, 3, 3}/2 shown in Figure 2, and hence we can think that P lives in the projective 3-space.…”