Erdős–Gyárfás power-of-two cycles problem

Show that every finite graph with minimum degree at least 3 contains a cycle whose length is a power of two (2^k with k ≥ 2).

Background

This classical graph theory problem asks for the existence of cycles with lengths that are powers of two under a minimum degree condition. While results exist for large minimum degree (dense settings), the general case is unresolved.

A resolution would contribute to structural cycle theory and degree-constrained extremal graph results.

References

While the question remains open, it was shown that the claim was true if the minimum degree of $G$ was sufficiently large; in fact in that case there is some large integer $\ell$ such that for every even integer $m\in [(\log\ell)8,\ell]$, $G$ contains a cycle of length $m$.

— Mathematical exploration and discovery at scale  (2511.02864 - Georgiev et al., 3 Nov 2025) in Subsection “Erdős–Gyárfás conjecture” (Section 4.28)

The conjecture is open; it is listed as Problem~64 on erdosproblems.com.

— 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 1, Introduction