Truncation symmetry type graphs

There are operations that transform a map M (an embedding of a graph on a surface) into another map in the same surface, modifying its structure and consequently its set of flags F(M). For instance, by truncating all the vertices of a map M, each flag in F(M) is divided into three flags of the truncated map. Orbanic, Pellicer and Weiss studied the truncation of k-orbit maps for k < 4. They introduced the notion of T-compatible maps in order to give a necessary condition for a truncation of a k-orbit map to be either k-, 3k/2- or 3k-orbit map. Using a similar notion, by introducing an appropriate partition on the set of flags of the maps, we extend the results on truncation of k-orbit maps for k < 8 and k=9

The Graphicahedron

The paper describes a construction of abstract polytopes from Cayley graphs of symmetric groups. Given any connected graph G with p vertices and q edges, we associate with G a Cayley graph of the symmetric group S_p and then construct a vertex-transitive simple polytope of rank q, called the graphicahedron, whose 1-skeleton (edge graph) is the Cayley graph. The graphicahedron of a graph G is a generalization of the well-known permutahedron; the latter is obtained when the graph is a path. We also discuss symmetry properties of the graphicahedron and determine its structure when G is small.Comment: 21 pages (European Journal of Combinatorics, to appear

Embeddings of biplanes

Given a combinatorial design $D$ and its incidence graph $I_D$, the embedding of $D$ on a surface $S$ is defined by a transformation on the embedding of the graph $I_D$ on $S$. In this talk will be introduce this concepts to describe the embedding of the possible bipalnes known so far.Non UBCUnreviewedAuthor affiliation: AnahuacFacult