Claims

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1992_05_01_brown: Brown's 1992 example of a vertex 5-critical graph with no critical edge, the case k = 5, r = 1; refereed in Discrete Math.

2002_12_01_jensen: Jensen's 2002 vertex-critical k-chromatic graphs, for every k at least 5, in which no set of fewer than m edges at one vertex is critical, giving the case r = 1 for every k at least 5; refereed in Discrete Math.

2002_12_01_lattanzio: Lattanzio's 2002 circulant families of vertex k-critical graphs with no critical edge whenever k - 1 is not prime, the case r = 1 for those k; refereed in Discrete Math.

2023_10_19_martinsson_steiner: Martinsson and Steiner prove that for every r there is k_0 such that every k at least k_0 has a vertex-critical k-chromatic graph with no critical set of at most r edges; refereed in Combin. Probab. Comput.

2025_08_12_skottova_steiner: Skottova and Steiner's 2025 preprint proves f_k(n) = Omega(n^(1/3)) for every fixed k at least 5, so suitable graphs exist for all k at least 5 and all r; no journal version, unreviewed.

2026_09_09_chan: Alex Chan's explicit 4-chromatic graph on 60 vertices with every vertex critical and no critical edge, the k = 4 case of Dirac's conjecture, public from 9 September 2026; Lean certificates not built by this corpus.

2026_09_10_kitamura: Kenta Kitamura's AI-assisted Lean 4 proof of 10 September 2026 that a 4-chromatic graph on 48 vertices has every vertex critical and no critical edge, the k = 4, r = 1 case of Dirac's conjecture; pending.

2026_09_11_kruer_kohlmeyer: Lean 4 proof certified by Conjectures.io in September 2026 that for every k at least 4 and r at least 1 some k-chromatic graph has every vertex critical and no critical set of at most r edges; the k = 4 case for r at least 2 is new.

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