Infinite Eulerian tessellations

✍ Scribed by J.L. Brenner; R.C. Lyndon

• Publisher: Elsevier Science
• Year: 1983
• Tongue: English
• Weight: 1000 KB
• Volume: 46
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(83)90245-5

✦ Synopsis

An Eulerian path in a graph G is a path 11" such that (l) ,rr traverses each edge of G exactly once in each direction, and (2) *r does not traverse any edge once in one direction and then immediately after in the other direction. A tess~.'llation T of the plane is Eulerian if its l-skeleton G admits an Eu!orian path. It is showtt that the three regular tessellations of the Euclidean plane are Eulerian. More generally, if T is a tessellation of the plane such that ~ach face has at least p sides and each vertex has degree (number of incident edges) at least q. where l/p+ l/q<~12, then, except possibly for the case p::3 and q =6, T is Eulerian. Let T* be the truncation of 7". If every vertex of T has degree 3, then T* is no~ Eulerian. If ever), vertex has degree 4, or degree at least 6. then T is Eulerian.