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Claim. Theorem 1.3 of M. Bonamy and T. J. Perrett, Gallai's path decomposition conjecture for graphs of small maximum degree, Discrete Math. 342 (2019), no. 5, 1293--1299, first posted as arXiv:1609.06257v1 on 20 September 2016 (the claim's date): every connected graph on $n$ vertices with maximum degree at most $5$ admits a path decomposition into $\lceil n/2\rceil$ paths. The theorem is recorded, from the arXiv v1, on the theorem page of the source card; the journal text was not compared with it.

Covers. The statement of Problem 583 for connected graphs of maximum degree at most $5$.

Depends on. Nothing in this wiki.

Acceptance. Refereed: the paper is a publication in Discrete Mathematics. The site's curator credits the result while labeling the problem FALSIFIABLE, which is commentary on an open problem and not reviewed evidence.