Planar embeddings with infinite faces

✍ Scribed by C. Paul Bonnington; Mark E. Watkins

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 173 KB
• Volume: 42
• Category: Article
• ISSN: 0364-9024
• DOI: 10.1002/jgt.10089

✦ Synopsis

Abstract

We investigate vertex‐transitive graphs that admit planar embeddings having infinite faces, i.e., faces whose boundary is a double ray. In the case of graphs with connectivity exactly 2, we present examples wherein no face is finite. In particular, the planar embeddings of the Cartesian product of the r‐valent tree with K~2~ are comprehensively studied and enumerated, as are the automorphisms of the resulting maps, and it is shown for r = 3 that no vertex‐transitive group of graph automorphisms is extendable to a group of homeomorphisms of the plane. We present all known families of infinite, locally finite, vertex‐transitive graphs of connectivity 3 and an infinite family of 4‐connected graphs that admit planar embeddings wherein each vertex is incident with an infinite face. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 257–275, 2003