| import Mathlib | |
| namespace OAI | |
| namespace LinearListHadwiger | |
| def ListColorable {V : Type} (G : SimpleGraph V) {Color : Type} | |
| (L : V → Finset Color) : Prop := | |
| ∃ c : V → Color, (∀ v, c v ∈ L v) ∧ ∀ ⦃v w⦄, G.Adj v w → c v ≠ c w | |
| def Choosable {V : Type} (G : SimpleGraph V) (k : ℕ) : Prop := | |
| ∀ (Color : Type) (L : V → Finset Color), (∀ v, k ≤ (L v).card) → ListColorable G L | |
| noncomputable def listChromaticNumber {V : Type} (G : SimpleGraph V) : ℕ := | |
| sInf {k : ℕ | Choosable G k} | |
| def HasCliqueMinor {V : Type} (G : SimpleGraph V) (t : ℕ) : Prop := | |
| ∃ B : Fin t → Set V, | |
| (∀ i, (G.induce (B i)).Connected) ∧ | |
| (∀ i j, i ≠ j → Disjoint (B i) (B j)) ∧ | |
| (∀ i j, i ≠ j → ∃ v ∈ B i, ∃ w ∈ B j, G.Adj v w) | |
| noncomputable def hadwigerNumber {V : Type} (G : SimpleGraph V) : ℕ := | |
| sSup {t : ℕ | HasCliqueMinor G t} | |
| def MainStatement : Prop := | |
| ∃ C : ℕ, 1 ≤ C ∧ ∀ (V : Type) [Fintype V] [Nonempty V] (G : SimpleGraph V), | |
| listChromaticNumber G ≤ C * hadwigerNumber G | |
| end LinearListHadwiger | |
| end OAI | |