Cycle spectra, automorphic uniqueness, and isomorphism classes of generalized Petersen graphs

This study began with a focus on cycle sizes in the Generalized Petersen Graphs. A search of literature revealed very complete results on the existence and non-existence of hamiltonian cycles in the Generalized Petersen Graphs but little else. Focusing first on the Generalized Petersen Graphs P(m, k) that are modeled from the Petersen Graph itself where m and k are relatively prime; I settled completely the cycle spectrum (cycle lengths) of P( m, 2) for every odd m. My approach is completely constructive. For every cycle size located, I give an algorithm. Turning to P(m, 3) for m relatively prime to 3, I first noted that P(m, 3) and in fact P(m, k), for every m even and k odd, is bipartite which gives a different flavor to the cycle spectrum problem. For P(m, 3) I again characterize completely the cycle spectrum constructively.;Concerning P(m, k) with m relatively prime to k, I was able to characterize the girth and odd girth, that is the length of the shortest cycle and the length of the shortest odd cycle, respectively.;In the course of my investigation of these cycle questions. I noted that for a fixed m, it often happens that P( m, k) ≅ P(m, l). I was able to prove that in addition to l = m - k, P( m, k) ≅ P(m, l) if kl ≡ ±1 mod m. This result I later learned is in the literature but does not seem to be well-known.;Subsequently I was able to prove that in fact P( m, k) ≅ P(m, l) if and only if one of the following happens: (1) l =k; (2) l = m - k; (3) kl ≡ l mod m; (4)  kl ≡ -1 mod m..;As a consequence of this I was able to present a formula for the number of isomorphism classes of P (m, k). Prior to my work it was known that most Generalized Petersen Graphs are hamiltonian. In the course of my work on cycle spectra I noticed that in some cases P(m, k) is "Automorphically Uniquely Hamiltonian" that is to say any hamiltonian cycle is transformable into any other hamiltonian cycle by an automorphism. P( m, 2) has been characterized completely and I have a discussion of the beginning of the study of P(m, 3) but it remains open.;Numerous questions remain and are discussed at the end of this work.