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e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | import OAI.Combinatorics.HadwigerMatching.MinorBound
import OAI.Combinatorics.HadwigerMatching.ChromaticBound
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
namespace OAI
namespace HadwigerMatching
/-- The matching–minor proposition: the Hadwiger bound and its chromatic
consequence when the independence and touching-matching numbers are small. -/
theorem full_matching_minor_proposition
(V : Type) [Fintype V] [Nonempty V] (G : SimpleGraph V) :
let m : ℕ := Fintype.card V
let h : ℕ := LinearListHadwiger.hadwigerNumber G
let c : ℕ := HadwigerMatching.connectedMatchingNumber G
(h : ℚ) ≤ ((m : ℚ) + 4 * (c : ℚ) + 2) / 3 ∧
(G.indepNum ≤ 2 →
(c : ℚ) < (m : ℚ) / 100 →
5 ≤ m →
(h : ℚ) < 26 * (m : ℚ) / 75 + 2 / 3 ∧
26 * (m : ℚ) / 75 + 2 / 3 < (m : ℚ) / 2 ∧
(m : ℚ) / 2 ≤ (G.chromaticNumber.toNat : ℚ)) := by
dsimp only
have hn := hadwiger_matching_bound G
have hq : 3 * (LinearListHadwiger.hadwigerNumber G : ℚ) ≤
(Fintype.card V : ℚ) + 4 * (connectedMatchingNumber G : ℚ) + 2 := by
exact_mod_cast hn
refine ⟨by linarith, ?_⟩
intro hα hc hm
have hm' : (5 : ℚ) ≤ (Fintype.card V : ℚ) := by exact_mod_cast hm
have hnχ := card_le_twice_chromaticNumber_of_indepNum G hα
have hχ : (Fintype.card V : ℚ) ≤ 2 * (G.chromaticNumber.toNat : ℚ) := by
exact_mod_cast hnχ
refine ⟨?_, ?_, ?_⟩ <;> linarith
end HadwigerMatching
end OAI
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