Two-attachment gadget inequality conjecture

Prove or refute that every two-attachment gadget with no internal cycle whose length is a power of two satisfies $\max S\ge 2\min S$, where $S$ is the set of lengths of simple paths between the two attachments.

Background

The paper investigates two-attachment gadgets as a possible route to constructing graphs with no power-of-two cycles. If such a gadget had a path-length set SS satisfying max⁡S<2min⁡S\max S<2\min S, for example S={5,7}S=\{5,7\}, then splicing it into every edge of a suitable base graph might avoid all power-of-two cycle lengths.

The exhaustive gadget census found that every two-attachment gadget examined so far contains a 4-cycle or an 8-cycle. This motivates the stated conjecture, which remains unresolved and would rule out the proposed short-spectrum gadget strategy if true.

References

Every such gadget found so far contains a 4- or 8-cycle. We conjecture that a two-attachment gadget without internal power-of-two cycles always has $\max S\ge 2\min S$.

— Small graphs without power-of-two cycles: a lower bound of 24, a correction to a construction of Exoo, and explicit bounds for f(k)  (2609.04686 - Garcia, 4 Sep 2026) in Section 6, Optimality of 78 and gadget censuses