The Gray graph is pseudo 2-factor isomorphic

TL;DR

This work advances the study of pseudo $2$-factor isomorphic cubic bipartite graphs by establishing that the Gray graph on $54$ vertices with girth $8$ is another counterexample to the conjecture that only a few graphs admit this property. Building on Goedgebeur’s $30$-vertex counterexample, the authors perform extensive computer searches, extending exhaustive generation to order $42$ (girth $\ge6$) and order $52$ (girth $\ge8$), and finding no additional counterexamples beyond the known ones; the Gray graph is the smallest girth-$8$ counterexample. They also systematically examine censuses of highly symmetric graphs and implement robust verification tools, confirming the existing conjectures for broader graph families up to substantial orders and ensuring reproducibility through cross-checks. The results underscore the rarity of counterexamples and highlight open questions about possible infinite families and higher-girth instances, while validating related conjectures (e.g., the $2$-factor Hamiltonian conjecture) for specific regimes. Overall, the paper combines structural analysis with large-scale computation to map the landscape of pseudo $2$-factor isomorphic graphs and clarifies the boundary between known counterexamples and open problems in the area.

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.

Figures (6)

• Figure 1: The Heawood graph on the left and the Pappus graph on the right.
• Figure 2: The pseudo 2-factor isomorphic graph $\mathcal{G}$ on 30 vertices.
• Figure 3: The counterexample $\mathcal{G}$ on 30 vertices, with its 2--factor types highlighted.
• Figure 4: The counterexample $\mathcal{G}$ on 30 vertices, as constructed in AFLR.
• Figure 5: The Gray graph $\mathcal{GR}$.

Theorems & Definitions (7)

• Conjecture 1.1
• Conjecture 1.2: FJLS
• Conjecture 1.3
• Conjecture 1.4
• Corollary 3.3
• Corollary 3.4
• Corollary 3.6