A Mechanism Dichotomy for Power-of-Two Cycles in Honeycomb Toroidal Graphs of Girth 6

We prove that the Erdos–Gyárfás as conjecture holds for all honeycomb toroidal graphs (HTGs) of girth 6, establishing that the smallest exponent k such that Gcontains a cycle of length 2k satisfies kmin(G) ≤4. Our main tool is a novel Mechanism Dichotomy framework for cubic bipartite graphs of girth 6: Case 1: Tight pairs of hexagons exist ⇒8-cycles via symmetric difference (Thetaresonance). Case 2a: No tight pairs, but 8-cycles exist independently. Case 2b: No tight pairs and no 8-cycles ⇒16-cycles via Alspach’s bypass construction. For HTGs, we fully classify these cases using column-span analysis and Alspach’s cycle spectrum results.