A (k, g)-cage is a k-regular graph of girth g of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.
We present a new construction of infinite families of (finite as well as infinite) vertex-transitive graphs that are not Cayley graphs; many of these turn out even to be arc-transitive. The construction based on representing vertex-transitive graphs as coset graphs of groups, and on a simple but powerful necessary arithmetic condition for Cayley graphs.… (More)
The main topic of the paper is the question of the existence of self-complementary Cayley graphs Cay(G; S) with the property S 6 = G # n S for all 2 Aut(G). We answer this question in the positive by constructing an innnite family of self-complementary circulants with this property. Moreover, we obtain a complete classiication of primes p for which there… (More)
The automorphism groups Aut(C(G, X)) and Aut(CM(G, X, p)) of a Cayley graph C(G, X) and a Cayley map CM(G, X, p) both contain an isomorphic copy of the underlying group G acting via left translations. In our paper, we show that both automorphism groups are rotary extensions of the group G by the stabilizer subgroup of the vertex 1 G. We use this description… (More)
A regular Cayley map for a finite group A is an orientable map whose orientation-preserving automorphism group G acts regularly on the directed edge set and has a subgroup isomorphic to A that acts regularly on the vertex set. This paper considers the problem of determining which abelian groups have regular Cayley maps. The analysis is purely algebraic,… (More)