Power-of-two cycles in graphs covered by pentagons

We prove two special cases of the Erdős–Gyárfás conjecture. Every finite simple graph of minimum degree at least three with a spanning factor of pentagons contains a simple cycle whose length is a power of two. For cubic graphs, it suffices that every vertex lie on a pentagon, allowing overlap. The preprint includes proofs, a finite verification appendix, and a reproducible Python checker in the source archive. The manuscript documents the use of AI in its preparation.