Power-of-Two Cycles in Cubic Bipartite Vertex-Transitive Graphs of Girth Six

We prove that every cubic bipartite vertex-transitive graph of girth six contains a simplecycle whose length is a power of two, specifically one of 8, 16, or 32. This confirms theErdos-Gyárfás conjecture for this highly symmetric class with the sharp bound kmin ≤ 5.Our proof combines the structural classification of Poto£nik and Vidali, which partitionsthese graphs into four families, with a novel port voltage framework that encodes the interactionbetween ring structures and matching edges. For the toroidal family, local hexagoncombinatorics yields cycles of length 8 or 16. For the hyperbolic and dihedral truncationfamilies, we establish two independent obstructions: a local corner-cost bound arising fromantipodal port geometry that excludes 8-cycles, and a global Z2-holonomy obstruction thatexcludes 16-cycles. Despite these barriers, we prove that a canonical ground-state walk inthe quotient graph lifts to a simple 32-cycle.Computational verification over the complete census of cubic vertex-transitive graphs upto 1280 vertices confirms the theorem and identifies exactly 14 extremal graphs attainingkmin = 5.