{"data":[{"id":"kill-line-graph-signature-2026","conjecture_oneline":"The signature of a connected line graph is at most 1 (Akbari, Elphick, Kumar, Pragada, Tang, Discrete Math. 349 (2026) 114953, Conjecture 4.12).","resolution_type":"counterexample","resolvers":"Luke Francis and Trevor Uptain","date":"2026-07-24","evidence_url":null,"evidence_note":"14-vertex and 48-vertex counterexamples plus a proof that the signature of connected line graphs is unbounded (signature k+1 on 14k vertices). arXiv submission 2026-07-24; identifier pending announcement.","method_note":"Independent AI-assisted searches (ChatGPT 5.6 Pro; Claude Fable 5) converging on the same two-pentagons-and-a-square mechanism; all computations verified in exact rational arithmetic.","state":"confirmed"},{"id":"kill-dinitz-garg-goemans-2026","conjecture_oneline":"The Dinitz-Garg-Goemans conjecture on unsplittable flows (open roughly 30 years).","resolution_type":"counterexample","resolvers":"Dmitry Rybin","date":"2026-07-22","evidence_url":"https://x.com/DmitryRybin1/status/2079904005652893709","evidence_note":"Counterexample announced 2026-07-22, found with GPT-5.6 Pro; evidence link is the public announcement pending a posted preprint.","method_note":"Part of the July 2026 wave of AI-assisted refutations.","state":"confirmed"},{"id":"kill-txgraffiti-saturation-harmonic-2026","conjecture_oneline":"For every nontrivial connected graph, the minimum maximal matching number is at most the harmonic index (TxGraffiti conjecture, 2023).","resolution_type":"counterexample","resolvers":"Bıyıkoğlu; smallest cases identified by Gupta","date":"2026-06-20","evidence_url":"https://arxiv.org/abs/2606.15761","evidence_note":"Refuted with unbounded violation ratio; the smallest counterexample is the friendship graph F4 on nine vertices, and the smallest tree counterexample is the subdivided star on eleven vertices.","method_note":null,"state":"confirmed"},{"id":"kill-brouwer-2026","conjecture_oneline":"Brouwer's conjecture: the sum of the t largest Laplacian eigenvalues of any graph is at most e(G) + t(t+1)/2.","resolution_type":"proof","resolvers":"Kothari and Tudose","date":"2026-06-10","evidence_url":"https://arxiv.org/abs/2606.12197","evidence_note":"Full proof via an equivalence with the Grone-Merris-Bai theorem; cited as confirmed by subsequent July 2026 literature.","method_note":"A reminder that conjectures leave the open list in both directions: this one was proved, not refuted.","state":"confirmed"},{"id":"kill-inertia-quadratic-2026","conjecture_oneline":"Every graph satisfies 2n+(G) ≤ n−(G)(n−(G)+1) (Akbari, Elphick, Kumar, Pragada, Tang, quadratic inertia conjecture).","resolution_type":"counterexample","resolvers":"H. Chen and J. Li","date":"2026-05-15","evidence_url":"https://arxiv.org/abs/2605.07196","evidence_note":"Counterexamples to a conjecture on graph inertia; a dense family also refuting a related order inequality from the same paper.","method_note":null,"state":"confirmed"}],"notice":"Status is community- and machine-tracked and may lag. Verify independently before investing effort."}