The Gray graph is pseudo 2-factor isomorphic

2025-04-16

AI Summary

This work investigates whether the Gray graph serves as a counterexample to the pseudo-2-factor-isomorphic graph conjecture and systematically searches for smaller-order potential counterexamples. Method: Leveraging high-performance graph enumeration, constraint-based search, isomorphism pruning, and a novel algorithm for determining the parity of 2-factors—cross-validated against symmetric graph databases—we exhaustively analyze cubic bipartite graphs with girth ≥ 8. Results: We establish, for the first time, that the 54-vertex, girth-8 Gray graph is the second known counterexample to the conjecture, thereby refuting Abreu et al.’s claim that the only essentially 4-edge-connected pseudo-2-factor-isomorphic cubic bipartite graphs are $K_{3,3}$, the Heawood graph, and the Pappus graph. We rigorously prove the nonexistence of smaller-order counterexamples with girth 8. Furthermore, we extend verification of the 2-factor Hamiltonian conjecture to all cubic bipartite graphs of order ≤ 52 (girth ≥ 8) and ≤ 42 (arbitrary girth).

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Applications of cryptography

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.

Problem

• Disprove conjecture on pseudo 2-factor isomorphic graphs
• Identify Gray graph as a counterexample to the conjecture
• Verify 2-factor Hamiltonian conjecture for specific graph classes

Innovation

• Computer search for graph counterexamples
• Verification of 2-factor Hamiltonian conjecture
• Analysis of symmetrical graphs censuses