Is it true that every graph whose vertices have odd degree greater than one contains a cycle of length 2^(n) for some n

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G=(V,E)$ be a finite simple undirected graph, with $V\ne\varnothing$. Its vertex degrees are $\deg_G(v)=|\{w:\{v,w\}\in E\}|$. A cycle of length $\ell\geq3$ consists of distinct vertices $v_1,\ldots,v_\ell$ with edges $v_iv_{i+1}$, interpreting $v_{\ell+1}=v_1$.

Statement recorded in the supplied source.

If every $\deg_G(v)$ is an odd integer greater than one, must $G$ contain a cycle of length $2^r$ for some integer $r\geq2$? This is the recorded odd-degree instance of the Erdős--Gyárfás power-of-two cycle problem; the cited source states the broader minimum-degree-at-least-three conjecture. [S2]

Short English statement

Must every finite graph with all vertex degrees odd and at least three contain a cycle whose length is a power of two?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3100012 (AMR-030-0012). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Benny Sudakov and Jacques Verstraete, Cycle Lengths in Sparse Graphs (2007), abstract and section 1.3, pp. 1 and 3--4. https://arxiv.org/abs/0707.2117.