{"data":{"id":"min-order-line-graph-signature-2026","statement_oneline":"Determine the minimum order of a connected graph whose line graph has signature t, for each t at least 2.","statement_full_latex":"For each $t \\ge 2$, determine the minimum number of vertices of a connected graph $G$ with $s(L(G)) = t$. For $t = 2$: a 14-vertex example exists and is the unique subcubic example on at most 14 vertices, but no minimality claim over all connected graphs is known.","source":{"title":"The signature of connected line graphs is unbounded (Francis-Uptain)","arxiv":"pending","url":null,"year":2026,"area":"math.CO"},"renown":{"score":1,"rationale":"Posed July 2026 in the paper refuting the line-graph signature conjecture of Akbari, Elphick, Kumar, Pragada and Tang."},"attackability":{"score":4,"rationale":"Days old, essentially unhunted. Oracle is a single eigendecomposition; exhaustive search is complete for subcubic graphs through 14 vertices, leaving higher-degree small graphs and all t >= 3 wide open; the known 14-vertex and 48-vertex examples and the bridge/chain machinery are ready-made seeds.","subscores":{"finite_witness":5,"oracle_cost":5,"freshness":5,"seedability":4}},"status":"open","last_verified_open":"2026-07-24","added":"2026-07-24","effective_status":"open","claims":[],"status_events":[{"id":"seed-min-order-line-graph-signature-2026","conjecture_id":"min-order-line-graph-signature-2026","status":"open","date":"2026-07-24T00:00:00.000Z","note":"Loaded from seed data."}],"resolutions":[]},"notice":"Status is community- and machine-tracked and may lag. Verify independently before investing effort."}