Elementary Proof of the Closed Walk Stacking Criterion

Definitions and mathematical statement

Let $G$ be a graph and let $W$ be a closed walk in $G$. The notation $kW'$ in the supplied statement denotes repetition of a walk $W'$, traversing it $k$ times consecutively. The supplied record does not specify when this repetition is defined or which convention identifies two closed walks.

The requested task, identified as Problem 3 referring to Theorem 2, is to find an elementary proof of the criterion \[ W\text{ has a stacking in }G \quad\Longleftrightarrow\quad W\ne kW'\text{ for every walk }W'\text{ in }G\text{ and every integer }k\ge 2. \] The target is a proof request for the stated characterization, rather than a request to establish a newly conjectured characterization.

Unresolved formulation qualifications. The supplied record does not define a stacking. Consequently, the left-hand side is retained as a source term and the display is a transcription of the intended criterion, not a fully specified mathematical proposition. The record also leaves unspecified the permitted class of graphs, whether walks must satisfy a reducedness or immersion hypothesis, whether length-zero closed walks are allowed, whether equality is based or taken up to cyclic reparametrization, and whether the repetition quantifier is intended to range only over closed walks. These conventions and any necessary hypotheses must be recovered from the cited theorem before a precise standalone formulation can be accepted. The intended meaning of an elementary proof is likewise unspecified.

Short English statement

Find an elementary proof that a closed walk admits a stacking exactly when it is not a repetition of a shorter walk. The supplied record omits the definition of stacking and several walk conventions, so the precise scope requires review. Its supplied open label concerns the elementary-proof request.

Sources

• Local registry: supplied record from data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005111$, exact source code catalogue entry, title Elementary Proof of the Closed Walk Stacking Criterion. The record identifies Problem 3 and its reference to Theorem 2; its proposer field is empty and its proposed year is 2022.
• Supplied report citation: Combinatorics, Probability and Computing (2023), Oberwolfach Reports, pp. 1165--1237, DOI 10.4171/owr/2022/22. https://doi.org/10.4171/owr/2022/22. The extracted text includes the page marker 1226 and identifies Oberwolfach Report 22/2022.
• Supplied literature source, labelled "official OWR report": https://publications.mfo.de/bitstream/handle/mfo/3964/OWR_2022_22.pdf?isAllowed=y&sequence=4.
• Literature assessment dated exactly 2026-08-22: the supplied curation attributes the underlying stacking characterization to Louder and Witten and reports that no elementary-proof resolution was identified, labelling the proof request open. This is supplied dated curation, not independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified and says the official report was checked directly and extraction artifacts removed. These are supplied verification claims; no independent source consultation is asserted here.