Pseudo and strongly pseudo 2-factor isomorphic regular graphs and digraphs

A graph G is pseudo 2–factor isomorphic if the parity of the number of cycles in a 2–factor is the same for all 2–factors ofG. In [3] we proved that pseudo 2–factor isomorphic k–regular bipartite graphs exist only for k ≤ 3. In this paper we generalize this result for regular graphs which are not necessarily bipartite. We also introduce strongly pseudo 2–factor isomorphic graphs and we prove that pseudo and strongly pseudo 2–factor isomorphic 2k–regular graphs and k–regular digraphs do not exist for k ≥ 4. Moreover, we present constructions of infinite families of regular graphs in these classes. In particular we show that the family of Flower snarks is strongly pseudo 2–factor isomorphic but not 2–factor isomorphic and we conjecture that, together with the ∗This research was financially supported by the Engineering Faculty of Taranto of the Technical University of Bari (Politecnico di Bari), using funds of the Provincia di Taranto for the support of the faculty’s teaching and scientific activities.