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Claim. Theorem 1.3 of Y. Chu, G. Fan and C. Zhou, Gallai's conjecture and the path number of odd semi-cliques, Discrete Math. 349 (2026), Paper No. 114725, published online on 18 August 2025 (the claim's date; no preprint was found): a graph on $n$ vertices whose vertices of even degree induce $K_m$ with $m\le15$ has a path decomposition into at most $\lfloor n/2\rfloor+1$ paths. The authors observe (p. 2) that for odd $n$ this is $\lceil n/2\rceil$, the conjectured bound. No connectedness is assumed. The theorem follows from the paper's Theorem 1.4 on stars whose removal leaves at most one even-degree vertex. The paper's Theorem 1.7, at most $(4n+6)/7$ paths for every semi-clique, adds no instance of the conjecture: it gives $\lceil n/2\rceil$ only for $n\le7$, and the semi-cliques on at most $7$ vertices have even-degree vertices inducing a complete graph on at most $7$ vertices, which Theorem 1.3 already covers. The theorems are recorded on the Theorem 1.3 page and the Theorem 1.7 page of the source card.

Covers. The statement of Problem 583 for connected graphs on an odd number $n$ of vertices whose even-degree vertices induce $K_m$ with $m\le15$.

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.