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

Show that every finite graph with minimum degree 3 has a cycle of length 4, 8, 16 or another power of two, or find a counterexample.

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1	# Cycles of length a power of two (the Erdős–Gyárfás conjecture)
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3	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.
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5	## Problem
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7	Does every finite simple graph with minimum degree at least $3$ contain a cycle of length $2^k$ for some $k\ge2$?
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9	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.
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11	## Background
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13	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.
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15	## What is known
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17	- 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.
18	- 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.
19	- 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 $3$.
20	- 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.
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22	## What would count as progress
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24	- 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.
25	- Exhaustive searches that extend the known lower bounds on the size of a counterexample, with code and certificates.
26	- A proof for new classes, for example all cubic planar graphs or all cubic graphs.
27	- A formal proof against the Lean statement in [formal-conjectures](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/64.lean).
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29	## References
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31	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](https://doi.org/10.1090/jams/1018)
32	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](https://doi.org/10.37236/3252)
33	3. Y. Gao and S. Shan, Erdős–Gyárfás conjecture for $P_8$-free graphs (2021). [arXiv:2109.01277](https://arxiv.org/abs/2109.01277)
34	4. Z. Hu and C. Shen, Erdős–Gyárfás conjecture for $P_{10}$-free graphs (2023). [arXiv:2308.05675](https://arxiv.org/abs/2308.05675)
35	5. K. Markström, Extremal graphs for some problems on cycles in graphs (2004). [PDF](http://abel.math.umu.se/~klasm/Uppsatser/cycex.pdf)
36	6. A. Carr, Cycles of length 4 or 8 in graphs with diameter 2 and minimum degree at least 3 (2025). [arXiv:2508.19302](https://arxiv.org/abs/2508.19302)
37	7. G. Ducoffe and B. Dumitru, Towards a more structured search for Erdős–Gyárfás counter-examples (2026). [arXiv:2609.28594](https://arxiv.org/abs/2609.28594)
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39	## Source
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41	Imported from [Erdős Problems 64](https://www.erdosproblems.com/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.