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.

翻译：如果一个图的所有2-因子具有相同奇偶性的环数，则称该图为伪2-因子同构图。Abreu等人[J. Comb. Theory, Ser. B. 98 (2008) 432--442]猜想$K_{3,3}$、希伍德图和帕普斯图是仅有的本质4-边连通伪2-因子同构三次二部图。Goedgebeur [Discr. Appl. Math. 193 (2015) 57--60]通过构造一个30个顶点的反例$\mathcal{G}$（围长为6）否定了这一猜想。利用计算机搜索，他还证明该反例在至少40个顶点内是唯一的，且在至少48个顶点内不存在围长大于6的反例。本文证明了拥有54个顶点和围长8的格雷图同样是伪2-因子同构图猜想的反例。除图$\mathcal{G}$之外，这是已知的唯一其他反例。通过计算机搜索，我们证明不存在更小的围长为8的反例，且在所有围长下至少42个顶点内没有其他反例。此外，我们验证了已知对称图目录中也不存在其他反例。回顾定义：如果一个图的所有2-因子都是哈密顿环，则称该图为2-因子哈密顿图。作为本文计算机搜索的副产品，我们验证了Funk等人[J. Comb. Theory, Ser. B. 87(1) (2003) 138--144]提出的尚未解决的2-因子哈密顿猜想，对于围长至少为8的三次二部图在52个顶点内成立，对于任意围长的图在42个顶点内成立。