Growth functions on Fuchsian groups and the Euler characteristic

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46618/1/222_2005_Article_BF01405088.pd

Infinite Eulerian tessellations

An Eulerian path in a graph G is a path [pi] such that (1) [pi] traverses each edge of G exactly once in each direction, and (2) [pi] does not traverse any edge once in one direction and then immediately after in the other direction. A tessellation T of the plane is Eulerian if its l-skeleton G admits an Eulerian path. It is shown that the three regular tessellations of the Euclidean plane are Eulerian. More generally, if T is a tessellation of the plane such that each face has 2t least p sides and each vertex has degree (number of incident edges) at least q, where , then, except possibly for the case p = 3 and Q = 6, T is Eulerian. Let T* be the truncation of T. If every vertex of T has degree 3, then T* is not Eulerian. If every vertex has degree 4, or degree at least 6, then T is Eulerian.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25369/1/0000818.pd