A counterexample to the pseudo 2-factor isomorphic graph conjecture

TL;DR

This paper disproves a conjecture about pseudo 2-factor isomorphic cubic bipartite graphs by providing a counterexample with 30 vertices and shows this is the only such counterexample up to 40 vertices.

Contribution

The authors construct the first known counterexample to the conjecture and verify the uniqueness of such counterexamples up to 40 vertices.

Findings

Counterexample with 30 vertices disproves the conjecture.

This is the only counterexample up to at least 40 vertices.

The conjecture about 2-factor hamiltonian cubic bipartite graphs holds up to 40 vertices.

Abstract

A graph $G$ is pseudo 2-factor isomorphic if the parity of the number of cycles in a 2-factor is the same for all 2-factors of $G$. Abreu et al. 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 (Abreu et al., Journal of Combinatorial Theory, Series B, 2008, Conjecture 3.6). Using a computer search we show that this conjecture is false by constructing a counterexample with 30 vertices. We also show that this is the only counterexample up to at least 40 vertices. A graph $G$ is 2-factor hamiltonian if all 2-factors of $G$ are hamiltonian cycles. Funk et al. conjectured that every 2-factor hamiltonian cubic bipartite graph can be obtained from $K_{3,3}$ and the Heawood graph by applying repeated star products (Funk et al., Journal of Combinatorial Theory, Series B, 2003, Conjecture 3.2). We verify that this conjecture holds up to at least 40 vertices.