The planar cubic Cayley graphs of connectivity 2

Abstract

We classify the planar cubic Cayley graphs of connectivity 2, providing an explicit presentation and embedding for each of them. Combined with [9] this yields a complete description of all planar cubic Cayley graphs.

The Planar Cubic Cayley Graphs

“…These are the 1-ended ones, those of connectivity 1, and some of those appearing in Table 1 with connectivity 2 [13]. This paper is mainly concerned with the remaining ones.…”

“…Cayley graphs of connectivity 2 do not always admit a consistent embedding [9]. However, in the cubic case they do; see [13].…”

“…The above presentations are planar; see [13,Corollary 6.3]. We use Theorem 5.1 to obtain the entries (3)-(11) of Table 1, but we also use these graphs as important building blocks in later constructions of 3-connected Cayley graphs.…”

“…After some definitions and basic facts, we handle the finite and 1-ended case, corresponding to entries (12) to (19) of Table 1, in Section 7; entries (1) and (2) are easy and entries (3) to (11) were handled in [13]; see Section 5. Most of this paper is concerned with entries from (20) on.…”

“…The planar cubic Cayley graphs of connectivity 2 were completely analysed in [13], yielding the following classification.…”