A Proof of the Erdos-Gyárfás Conjecture

The Erdos-Gyárfás conjecture (1995) states that every graph with minimum degree atleast 3 contains a cycle whose length is a power of 2. We prove this conjecture in fullgenerality. The proof proceeds by case analysis on the girth: for girth 4, a 4-cycle exists bydenition; for girth at least 5, we prove that an 8-cycle exists through a structural analysis ofthe second neighborhood (fold forcing). For girth 3, we demonstrate a dichotomy: either a4-cycle exists (Cases 17), or the local structure mimics high-girth graphs (Case 8), forcingan 8-cycle. We exhibit an n = 15 graph that satises Case 8 (no 4-cycles) but contains 8-cycles, validating this structural propagation. This reveals the deeper mechanism: minimumdegree 3 forces power-of-two cycles, but the specic power depends on whether the graphcan locally close at length 4 or must propagate tension to length 8.