Abreu–Diwan–Jackson–Labbate–Schwenk's constituent classification conjecture for pseudo 2-factor isomorphic graphs

The conjecture asserted that every essentially $4$-edge-connected pseudo $2$-factor isomorphic cubic bipartite graph is one of $K_{3,3}$, the Heawood graph, or the Pappus graph.

Known results

An essentially $4$-edge-connected example of girth $4$ must be $K_{3,3}$; the full classification was formulated as a conjecture in the earlier literature. Goedgebeur’s search found a $30$-vertex counterexample, the only one found up to at least $40$ vertices; no girth-$8$ counterexample was found up to at least $48$ vertices.

Further counterexample reported in 2025

A later source reports that the $54$-vertex Gray graph is also pseudo $2$-factor isomorphic, making it the only other known counterexample besides Goedgebeur’s graph. It likewise reports no additional examples up to at least $42$ vertices and describes the original and broader classification conjectures as false.

Current status (as of September 2026): The classification conjecture is refuted by Goedgebeur’s reported $30$-vertex counterexample, with the Gray graph providing a later additional example; no further exact classification is established here.