Polyhedral embeddings of large face-width where all faces have the same length - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T12:07:49Z http://mathoverflow.net/feeds/question/99727 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99727/polyhedral-embeddings-of-large-face-width-where-all-faces-have-the-same-length Polyhedral embeddings of large face-width where all faces have the same length becko 2012-06-15T17:20:47Z 2012-06-17T14:25:33Z <p>Where can I find examples of polyhedral embeddings of simple graph with large face-width, such that all the faces have the same length? </p> <p>By polyhedral embedding I mean an embedding of the graph on a closed 2-dimensional surface, where the boundaries of all the faces are simple cycles, and the intersection of any two faces is either empty, a vertex, or an edge.</p> <p>The examples should be as non-homogeneous as possible. For example, the <a href="http://en.wikipedia.org/wiki/Archimedean_solid" rel="nofollow">Archimedean solids</a> are very homogeneous objects, because all the vertices are equivalent (for any two vertices, there is an isometry that takes one vertex to the other). I admit that I am not using "non-homogeneous" in a clearly defined sense. I am just asking for examples that do not have obvious symmetries.</p> <p><strong>Note</strong> : The face-width of the embedding is the smallest integer \$k\$ such that there exist \$k\$ facial walks whose union contains a noncontractible cycle. A facial walk is a walk that bounds a face of the embedding. The length of a face is simply the length of its face walk. Since we are dealing with polyhedral embeddings, all face walks are simple cycles.</p>