The Erdos-Gyárfás Conjecture for Cubic Vertex-Transitive Bipartite Graphs of Girth Six: A Complete Census Verification with Structural Analysis

We present a complete computational verification of the Erdos-Gyárfás conjecture forall 58,438 cubic vertex-transitive bipartite graphs of girth 6 in the CVT census (up to 1280vertices). Every graph in this class contains a cycle whose length is a power of two, withthe exponent bounded by 5. We establish a dyadic trichotomy: graphs partition into threeclasses based on kmin(G), the smallest k ≥ 3 such that a 2k-cycle exists. The vast majority(55,556 graphs) have kmin = 4, while 2,868 achieve kmin = 3 (containing 8-cycles), andexactly 14 counterexample candidates require kmin = 5 (containing 32-cycles but no 8- or16-cycles). These 14 extremal cases are precisely the PV(b) and PV(c) truncations in thecensus. We prove computationally that C8(G) = C16(G) = ∅ for all 14 instances and provideexplicit 32-cycle witnesses. A structural explanation via canonical matching decomposition,port-geometry, and shift-balance constraints elucidates why these specific cycle lengths areobstructed.