Planar embeddings with infinite faces

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 ¼ 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 ------------------

The Planar Cubic Cayley Graphs

“…As a always reverses spin in this case (see (4) and Lemma 8.4), and our embeddings are consistent, it follows easily that any two face-boundaries of G can be mapped to each other by a colourautomorphism. Thus all face-boundaries are induced by that relator, and so they are finite.…”

Between ends and fibers

“…We also noted that translatable rays need not be metric. We will show that neither of these statements holds for facial double rays in a planar map, obtaining therefrom a generalization of Theorem 2.1 of [2] from fibers to bundles. z, τ(b)).…”

“…There is a P 1 Q 2 -path R 4 that meets P 1 at a vertex r 4,1 farther from 1 along P 1 than r 3,1 and meets Q 2 at the vertex r 4,2 farther along Q 2 from u than any vertex of 1 . We may specify that R 4 have no interior vertex in 2 3,3 , contrary to planarity. This concludes the initial step of the induction.…”

“…Let w be the vertex in 1 ∩ 2 nearest to p on 1 [u, p), and let z = p be the vertex in 1 ∩ 2 nearest to p in the subray of P 1 with origin p. The paths 1 [p, w] and 1 [p, z] are nontrivial with no interior vertex on 2 .…”

“…This facilitates the proof of the main result of this section, that any two rays whose union is the boundary of an infinite face of a locally finite, almost-transitive planar map belong to different bundles. In [2] it was shown that such rays belong merely to distinct fibers. Among the three conjectures posed in the concluding Section 8 is that such rays belong to distinct ends.…”