  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 of $G$. Abreu et al.
conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the only
essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs
(Abreu et al., Journal of Combinatorial Theory, Series B, 2008, Conjecture
3.6).
  Using a computer search we show that this conjecture is false by constructing
a counterexample with 30 vertices. We also show that this is the only
counterexample up to at least 40 vertices.
  A graph $G$ is 2-factor hamiltonian if all 2-factors of $G$ are hamiltonian
cycles. Funk et al. conjectured that every 2-factor hamiltonian cubic bipartite
graph can be obtained from $K_{3,3}$ and the Heawood graph by applying repeated
star products (Funk et al., Journal of Combinatorial Theory, Series B, 2003,
Conjecture 3.2). We verify that this conjecture holds up to at least 40
vertices.
