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Claim. The account herong, in a comment of 17 September 2026 (the claim's date) on the site's discussion thread, reports an independent decider for the statement of Problem 583 on small graphs. It prunes with a lower bound on the number of paths inside each component $C$ of the uncovered part, the largest of $1$, $\lceil\mathrm{odd}_C/2\rceil$, $\lceil\Delta_C/2\rceil$ and $\lceil m_C/(n_C-1)\rceil$, summed over the components; the $\lceil\Delta/2\rceil$ term is new (a simple path uses at most two edges at a vertex). Its exhaustive sweeps at $n=9$, $10$ and $11$ ($261{,}080$, $11{,}716{,}571$ and $1{,}006{,}700{,}565$ connected graphs, the shard counts checked against OEIS A001349) find no counterexample and nothing undecided. As a control, on all $261{,}080$ connected graphs of order $9$ with the path budget cut to $4$ and then to $3$, its decider and the search of the earlier comment by sallerk, fitted with the same bound, report identical sets of undecomposable graphs. The code and logs are in a public repository, linked above at its commit of 17 September 2026, and the comment discloses that the decider and the write-up were prepared with assistance from Claude (Anthropic).

Covers. Connected graphs on at most $11$ vertices. Nothing for $n\ge12$: the $n=12$ sweep ($164{,}059{,}830{,}476$ graphs) was reported as running, and no outcome was posted.

Depends on. Nothing in this wiki.

Standing. Claimed: a forum computation, which the site does not verify, independent of and agreeing with sallerk's check; not reviewed and given no credit by this corpus, which has not run it. The claim is partial, so the problem's standing is unchanged by it.