The Resonance Barrier Conjecture for Cubic Vertex-Transitive Graphs

We investigate the Resonance Barrier Conjecture (RBC), which asserts that for cubicbipartite vertex-transitive graphs G with girth 6, the absence of 8-cycles implies the existence of 16-cycles: C8 (G) = ∅ ⇒ C16 (G) ̸= ∅. Using the Poto£nikVidali classication of cubic vertex-transitive graphs of girth 6 combined with a complete sweep of the CVT census (111, 360 graphs up to n = 1280), we establish the following results. Main results: (1) For graphs in PV (a) generic (signature (2, 2, 2) ), we prove RBC unconditionally via a universal cover argument using the PV map structure. (2) RBC fails in the truncation cases PV (b) and PV (c): we identify exactly 14 counterexamples in the CVT census, all structurally localized with parameter ℓ ≥ 14. (3) The falsication is informative: no counterexample has ℓ ∈ 10, 12, suggesting a sharp threshold phenomenon. Our methodology combines theoretical proof via the PV map structure with exhaustive computational verication, demonstrating how classication theorems can reduce innite problems to nite case analysis.