2-Factor hamiltonian graphs

The Heawood graph and K3,3 have the property that all of their 2-factors are Hamilton circuits. We call such graphs 2-factor hamiltonian. We prove that if G is a k-regular bipartite 2-factor hamiltonian graph then either G is a circuit or k=3. Furthermore, we construct an infinite family of cubic bipartite 2-factor hamiltonian graphs based on the Heawood graph and K3,3 and conjecture that these are the only such graphs.