3 - TRIVALENT CAYLEY GRAPHS; THE LCF AND FRUCHT NOTATIONS

Abstract

This chapter discusses the trivalent Cayley graphs and the trivalent and connected vertex-transitive graphs. It also presents an introduction of a subdivision of the classes—S, T, and Z—in subclasses or types, according to the possibility of obtaining the graphs under discussion as Cayley graphs. If it is impossible to obtain a given trivalent vertex-transitive graph as a Cayley graph, the prefix N is used. If a trivalent graph is the Cayley graph of some group H, there are only two possibilities as to the number of generators used in the construction of the Cayley diagram: (1) either to use one involutory generator and one of period ≤ 3 or (2) to use three involutory generators. The two cases are distinguished by using the presuperscript 1 or 3, respectively. It is not known whether Cayley graphs or at least 0-symmetric graphs are always hamiltonian. A convenient device for the concise description of a trivalent hamiltonian graph is called “LCF notation,” which was explained by Frucht.