Given a connected graph G with p vertices and q edges, the G-graphicahedron is a vertex-transitive simple abstract polytope of rank q whose edge-graph is isomorphic to a Cayley graph of the symmetric group Sp associated with G. The paper explores combinatorial symmetry properties of G-graphicahedra, focussing in particular on transitivity properties of their automorphism groups. We present a detailed analysis of the graphicahedra for the q-star graphs K1, q and the q-cycles Cq. The Cq-graphicahedron is intimately related to the geometry of the infinite Euclidean Coxeter group Ãq − 1 and can be viewed as an edge-transitive tessellation of the (q − 1)-torus by (q − 1)-dimensional permutahedra, obtained as a quotient, modulo the root lattice Aq − 1, of the Voronoi tiling for the dual root lattice Aq − 1 * in Euclidean (q − 1)-space.
Isabel Hubard, Universidad Nacional Autónoma de México.
Instituto de Matemáticas, professor
Deborah Oliveros, Universidad Nacional Autónoma de México.