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Confirmed proofs, counterexamples, and prior work that close Board entries.
| Conjecture | Resolution | Resolver | Date | Evidence |
|---|---|---|---|---|
| The signature of a connected line graph is at most 1 (Akbari, Elphick, Kumar, Pragada, Tang, Discrete Math. 349 (2026) 114953, Conjecture 4.12). | counterexample | Luke Francis and Trevor Uptain | arXiv link pending | |
| The Dinitz-Garg-Goemans conjecture on unsplittable flows (open roughly 30 years). | counterexample | Dmitry Rybin | evidence | |
| For every nontrivial connected graph, the minimum maximal matching number is at most the harmonic index (TxGraffiti conjecture, 2023). | counterexample | Bıyıkoğlu; smallest cases identified by Gupta | evidence | |
| Brouwer's conjecture: the sum of the t largest Laplacian eigenvalues of any graph is at most e(G) + t(t+1)/2. | proof | Kothari and Tudose | evidence | |
| Every graph satisfies 2n+(G) ≤ n−(G)(n−(G)+1) (Akbari, Elphick, Kumar, Pragada, Tang, quadratic inertia conjecture). | counterexample | H. Chen and J. Li | evidence |