Memoirs of the American Mathematical Society 1997; 75 pp; softcover Volume: 126 ISBN10: 0821805568 ISBN13: 9780821805565 List Price: US$42 Individual Members: US$25.20 Institutional Members: US$33.60 Order Code: MEMO/126/601
 The nine finite, planar, 3connected, edgetransitive graphs have been known and studied for many centuries. The infinite, locally finite, planar, 3connected, edgetransitive graphs can be classified according to the number of their ends (the supremum of the number of infinite components when a finite subgraph is deleted). Prior to this study the 1ended graphs in this class were identified by Grünbaum and Shephard as 1skeletons of tessellations of the hyperbolic plane; Watkins characterized the 2ended members. Any remaining graphs in this class must have uncountably many ends. In this work, infiniteended members of this class are shown to exist. A more detailed classification scheme in terms of the types of Petrie walks in the graphs in this class and the local structure of their automorphism groups is presented. Explicit constructions are devised for all of the graphs in most of the classes under this new classification. Also included are partial results toward the complete description of the graphs in the few remaining classes. Readership Graduate students and research mathematicians, chemists, and physicists interested in infinite repeating patterns. Table of Contents  Abstract
 Introduction
 Preliminaries
 Stabilizers
 Petrie walks
 Endsseparating circuits
 Plane graphs of circuit type
 Construction of plane graphs of mixed type
 Deconstruction of plane graphs of mixed type
 Ordinary graphs of mixed type
 Extraordinary graphs of line type
 Extraordinary graphs of mixed type
 Conclusions, conjectures and open questions
 Appendix A. Proof of Theorem 3.5
 Appendix B. Proof of Theorem 4.2
 References
