Cycles of length a power of two (the Erdős–Gyárfás conjecture)

Does every graph in which each vertex has at least three neighbours contain a cycle of length $4$, $8$, $16$ or some other power of two? Erdős and Gyárfás raised the question in the early 1990s and, unusually, expected a counterexample. Since then the statement has been proved in many special classes and for large minimum degree, but the original question for minimum degree three is open, and a single finite graph would settle it.

Problem

Does every finite simple graph with minimum degree at least $3$ contain a cycle of length $2^k$ for some $k\ge2$?

The conjecture is false for infinite graphs (an infinite tree with all degrees $3$ has no cycles at all), so finiteness matters. Erdős offered $1000 for this problem.

Background

The question is due to Erdős and Gyárfás. They believed the answer to be negative, and even that for every $r$ there are graphs of minimum degree at least $r$ with no cycle whose length is a power of two. In 1984 Erdős had asked whether large chromatic number or large average degree forces such a cycle, and Liu and Montgomery answered that question in a paper on cycle lengths in graphs, which also disproved the stronger expectation of Erdős and Gyárfás. The remaining difficulty is the case of small minimum degree, above all cubic graphs.

What is known

• Large degree. Liu and Montgomery (2023) proved that every graph of sufficiently large average degree contains a cycle of length $2^k$; indeed for some large $\ell$ it contains cycles of every even length in $[(\log\ell)^8,\ell]$. So the conjecture holds once the minimum degree exceeds an absolute constant.
• Special classes. The conjecture is known for $3$-connected cubic planar graphs (Heckman and Krakovski, 2013), for graphs with no induced path on $8$ vertices (Gao and Shan) and on $10$ vertices (Hu and Shen), for graphs of diameter $2$ (Carr, 2026), and for several other classes listed on the problem's forum.
• Computer searches. Markström (2004) showed that a cubic counterexample needs at least $30$ vertices. Ducoffe and Dumitru (2026, preprint) verified the conjecture for all graphs with at most $40$ vertices, all bipartite graphs with at most $66$ vertices and all cubic graphs with at most $48$ vertices, and showed that in a minimal counterexample more than $2/3$ of the vertices have degree $3$.
• Claims. A claimed proof of the full conjecture, a long case analysis with a partial Lean formalisation, was posted in August 2026 (G. Duran-Ballester, listed on the problem's proof-claims page). A gap was pointed out within days and a repair proposed; the proof has not been independently verified.

What would count as progress

• A counterexample: a finite graph with minimum degree at least $3$ and no cycle of length $4,8,16,\dots$, which can be verified by computer.
• Exhaustive searches that extend the known lower bounds on the size of a counterexample, with code and certificates.
• A proof for new classes, for example all cubic planar graphs or all cubic graphs.
• A formal proof against the Lean statement in formal-conjectures.

References

1. H. Liu and R. Montgomery, A solution to Erdős and Hajnal's odd cycle problem, J. Amer. Math. Soc. (2023). doi:10.1090/jams/1018
2. C. C. Heckman and R. Krakovski, Erdős–Gyárfás conjecture for cubic planar graphs, Electron. J. Combin. 20(2) (2013), P7. doi:10.37236/3252
3. Y. Gao and S. Shan, Erdős–Gyárfás conjecture for $P_8$-free graphs (2021). arXiv:2109.01277
4. Z. Hu and C. Shen, Erdős–Gyárfás conjecture for $P_{10}$-free graphs (2023). arXiv:2308.05675
5. K. Markström, Extremal graphs for some problems on cycles in graphs (2004). PDF
6. A. Carr, Cycles of length 4 or 8 in graphs with diameter 2 and minimum degree at least 3 (2025). arXiv:2508.19302
7. G. Ducoffe and B. Dumitru, Towards a more structured search for Erdős–Gyárfás counter-examples (2026). arXiv:2609.28594

Source

Imported from Erdős Problems 64, curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.