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.