| import OAI.Combinatorics.ListHadwiger.WeightedBound |
|
|
| namespace OAI |
|
|
| namespace LinearListHadwiger |
|
|
| lemma cliqueBound_hadwiger {V : Type} [Fintype V] (G : SimpleGraph V) : |
| ReedSeymour.CliqueBound G (hadwigerNumber G) := by |
| classical |
| intro I _ B hconn hdisj htouch |
| apply cliqueMinor_le_hadwigerNumber |
| let e := (Fintype.equivFin I).symm |
| refine ⟨fun i => B (e i),fun i => hconn (e i),?_,?_⟩ |
| · intro i j hij |
| exact hdisj _ _ (fun hh => hij (e.injective hh)) |
| · intro i j hij |
| exact htouch _ _ (fun hh => hij (e.injective hh)) |
|
|
| namespace Fractional |
| variable {V : Type} [Fintype V] [DecidableEq V] |
|
|
| lemma Dual.mass_le_one {V : Type} [Fintype V] [DecidableEq V] {G : SimpleGraph V} {w : V → ℝ} (hw : Dual G w) |
| {S : Set V} (hS : ReedSeymour.Independent G S) : ReedSeymour.mass w S ≤ 1 := by |
| classical |
| have hs : Stable G (Finset.univ.filter (· ∈ S)) := by |
| intro v hv z hz |
| exact hS v (Finset.mem_filter.mp hv).2 z (Finset.mem_filter.mp hz).2 |
| simpa [ReedSeymour.mass,Finset.sum_filter] using hw.2 _ hs |
|
|
| theorem number_le_two_hadwiger (G : SimpleGraph V) : |
| number G ≤ 2 * (hadwigerNumber G : ℝ) := by |
| classical |
| obtain ⟨w,hw,he⟩ := duality G |
| rw [← he] |
| cases isEmpty_or_nonempty V with |
| | inl h => |
| let := h |
| simp |
| | inr h => |
| let := h |
| exact ReedSeymour.dual_weight_bound w (by have hh := one_le_hadwigerNumber G; omega) |
| (cliqueBound_hadwiger G) (fun S hS => hw.mass_le_one hS) |
|
|
| theorem number_le_four_hadwiger (G : SimpleGraph V) : |
| number G ≤ 4 * (hadwigerNumber G : ℝ) := by |
| have hh := number_le_two_hadwiger G |
| have hp : (0 : ℝ) ≤ hadwigerNumber G := Nat.cast_nonneg _ |
| linarith |
| end Fractional |
|
|
| namespace SmallList |
| lemma no_fractionalObstruction {V : Type} [Fintype V] [DecidableEq V] |
| (G : SimpleGraph V) : ¬ FractionalObstruction G := by |
| rintro ⟨S,hS⟩ |
| exact (not_lt_of_ge (Fractional.number_le_four_hadwiger (G.induce (↑S : Set V)))) hS |
| end SmallList |
|
|
| theorem main_theorem : MainStatement := by |
| obtain ⟨C,hC,h⟩ := Recursion.final_bound_or_obstruction |
| refine ⟨C,hC,?_⟩ |
| intro V _ _ G |
| classical |
| exact (h V G).resolve_right (SmallList.no_fractionalObstruction G) |
|
|
| end LinearListHadwiger |
|
|
| end OAI |
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|
