Which generalized petersen graphs are cayley graphs?

Abstract

The generalized Petersen graph GP(n, k), n 2 3, 1 I k < n/2 is a cubic graph with vertex-set { u , ; i E Z,,} U { u J ;

Regular orientable embeddings of complete bipartite graphs

“…Besides the complete graphs, not much is known so far although regular embeddings of several infinite classes of graphs have been constructed. To the author's knowledge, the only graphs whose regular embeddings have been completely classified are the complete graph, the cocktail-party graph K 2n À nK 2 [11], the complete multipartite graph K p;p;...;p for p a prime [2] (see [13] for K p;p ), arc-transitive graphs of order a product of any two primes [3], a bouquet of circles [15], and a dipole [10]. As concerns other families of graphs, only partial results are known.…”

Regular Embeddings of Canonical Double Coverings of Graphs

“…It follows that the generalized Petersen graphs GP(n, k) with k 2 # &1 (mod n) provide a simple and interesting infinite family of cubic vertex-transitive graphs that are not Cayley graphs. It is not necessary to emphasize that the methods of Frucht, Graver and Watkins [4] are different from those employed in [15] and here.…”

“…The fact that G is connected is reflected by the transitive action of the permutation group (P, L), which we call the monodromy group of M and denote by Mon(M). (In [15] the same group is denoted by G(M).) This usually allows us to omit vertices from consideration whenever we use such a description of a map.…”

“…Theorem 9.1 has an interesting application to the generalized Petersen graphs, whose details are given in [15]. The generalized Petersen graph GP(n, k), n 3, 1 k<nÂ2, is a cubic graph with vertex-set [u i ; i # Z n ] _ [v i ; i # Z n ] and edge-set [u i u i+1 , u i v i , v i v i+k ; i # Z n ].…”

On the Spectrum of the Generalised Petersen Graphs

“…We will bound B(N ) crudely from above. First, we need the result of Nedela andŠkoviera [6] that P (n, k) is a Cayley graph if and only if k 2 ≡ 1 mod n. Therefore, we have that…”