Connectivity and planarity of Cayley graphs
A group \(G\) is said to be planar provided there exists a generating set \(X\) for \(G\) such that the Cayley graph \(C(G, X)\) is a planar graph. Although all finite planar groups have been classified by Maschke more than a hundred years ago, the general problem of the classification of all (infinite) planar groups seems to be considerably harder. One possible approach to the problem is to start the classification with groups possessing Cayley graphs of low connectivity. The connectivity \(\kappa(\Gamma)\) of a graph \(\Gamma\) is defined to be the size of the smallest separation set of \(\Gamma\), i.e., the size of the smallest vertex set of \(\Gamma\) whose removal separates \(\Gamma\) into two or more connected components. The connectivity \(\kappa(G)\) of a group \(G\) is then the minimum connectivity of all Cayley graphs of \(G\); while the planar connectivity \(\kappa_p(G)\) of a planar group \(G\) is the minimal connectivity of all planar Cayley graphs of \(G\). The main results of the paper include a classification of groups of connectivity 1: \(\kappa(G) =1\) if and only if \(G\) is either infinite cyclic or a nontrivial free product of infinite cyclic groups and groups of connectivity of two or more. Moreover, \(\kappa_p(G)=1\) implies for a planar \(G\) that \(G\) is either infinite cyclic or a free product of planar groups of planar connectivity of two or more. If \(\kappa (G)=2\), then \(G\) is either finite cyclic or dihedral, or it is the fundamental group of a graph of groups whose edge groups have order two or less and whose vertex groups have connectivity of at least three. If \(G\) admits a Cayley graph of connectivity 3 that is embeddable in a sphere, several results are presented that depend on the size of the set of accumulation points of vertices of the Cayley graph.
scientific article; zbMATH DE number 1208003