The Gray graph is pseudo 2-factor isomorphic

Abstract: A graph is pseudo 2-factor isomorphic if all of its 2-factors have the same parity of number of cycles. Abreu et al. [J. Comb. Theory, Ser. B. 98 (2008) 432--442] 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. This conjecture was disproved by Goedgebeur [Discr. Appl. Math. 193 (2015) 57--60] who constructed a counterexample $\mathcal{G}$ (of girth 6) on 30 vertices. Using a computer search, he also showed that this is the only counterexample up to at least 40 vertices and that there are no counterexamples of girth greater than 6 up to at least 48 vertices. In this manuscript, we show that the Gray graph -- which has 54 vertices and girth 8 -- is also a counterexample to the pseudo 2-factor isomorphic graph conjecture. Next to the graph $\mathcal{G}$, this is the only other known counterexample. Using a computer search, we show that there are no smaller counterexamples of girth 8 and show that there are no other counterexamples up to at least 42 vertices of any girth. Moreover, we also verified that there are no further counterexamples among the known censuses of symmetrical graphs. Recall that a graph is 2-factor Hamiltonian if all of its 2-factors are Hamiltonian cycles. As a by-product of the computer searches performed for this paper, we have verified that the $2$-factor Hamiltonian conjecture of Funk et al. [J. Comb. Theory, Ser. B. 87(1) (2003) 138--144], which is still open, holds for cubic bipartite graphs of girth at least 8 up to 52 vertices, and up to 42 vertices for any girth.