Irreducible pseudo 2-factor isomorphic cubic bipartite graphs

A bipartite graph is pseudo 2–factor isomorphic if all its 2–factors have the same parity of number of circuits. In a previous paper we have proved that pseudo 2–factor isomorphic k–regular bipartite graphs exist only for k ≤ 3, and partially characterized them. In particular we proved that the only essentially 4–edge-connected pseudo 2–factor isomorphic cubic bipartite graph of girth 4 is K3,3, and conjectured that the only essentially 4–edge-connected cubic bipartite graphs are K3,3, the Heawood graph and the Pappus graph. There is a characterization of symmetric v3 configurations due to Martinetti (1887) in which all v3 configurations can be obtained from an infinite set of so called irreducible configurations. Recall that v3 configurations correspond to cubic bipartite graphs with girth at least 6 via their Levi graphs. Recently the list of irreducible configurations has been completed by Boben (2007) in terms of their irreducible graphs. Here we use Boben’s list of irreducible graphs to prove that the only irreducible cubic bipartite graphs of girth at least 6 which are pseudo 2–factor isomorphic are Heawood and the Pappus graphs.