Pseudo 2-factor isomorphic regular bipartite graphs

A graph G is pseudo 2-factor isomorphic if the parity of the number of circuits in a 2-factor is the same for all 2-factors of G. We prove that there exist no pseudo 2-factor isomorphic k-regular bipartite graphs for k⩾4. We also propose a characterization for 3-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs and obtain some partial results towards our conjecture.