  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.
