**Memoirs of the American Mathematical Society**

1997;
75 pp;
Softcover

MSC: Primary 05;

Print ISBN: 978-0-8218-0556-5

Product Code: MEMO/126/601

List Price: $45.00

Individual Member Price: $27.00

**Electronic ISBN: 978-1-4704-0186-3
Product Code: MEMO/126/601.E**

List Price: $45.00

Individual Member Price: $27.00

# Locally Finite, Planar, Edge-Transitive Graphs

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*Jack E. Graver; Mark E. Watkins*

The nine finite, planar, 3-connected, edge-transitive graphs have been known and studied for many centuries. The infinite, locally finite, planar, 3-connected, edge-transitive 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 1-ended graphs in this class were identified by Grünbaum and Shephard as 1-skeletons of tessellations of the hyperbolic plane; Watkins characterized the 2-ended members. Any remaining graphs in this class must have uncountably many ends. In this work, infinite-ended 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.

#### Table of Contents

# Table of Contents

## Locally Finite, Planar, Edge-Transitive Graphs

- Contents v6 free
- Abstract vi7 free
- Chapter 1. Introduction 18 free
- Chapter 2. Preliminaries 512 free
- Chapter 3. Stabilizers 1118
- Chapter 4. Petrie Walks 2128
- Chapter 5. Ends-separating Circuits 2532
- Chapter 6. Plane Graphs of Circuit Type 2936
- Chapter 7. Construction of Plane Graphs of Mixed Type 3744
- Chapter 8. Deconstruction of Plane Graphs of Mixed Type 4552
- Chapter 9. Ordinary Graphs of Mixed Type 4956
- Chapter 10. Extraordinary Graphs of Line Type 5360
- Chapter 11. Extraordinary Graphs of Mixed Type 5966
- Chapter 12. Conclusions, Conjectures and Open Questions 6370
- Appendix A. Proof of Theorem 3.5 6774
- Appendix B. Proof of Theorem 4.2 7178
- References 7582

#### Readership

Graduate students and research mathematicians, chemists, and physicists interested in infinite repeating patterns.