| import OAI.Combinatorics.HadwigerMatching.SmallBranches | |
| import OAI.Combinatorics.HadwigerMatching.BranchGeometry | |
| import OAI.Combinatorics.HadwigerMatching.PairingExtension | |
| import OAI.Combinatorics.HadwigerMatching.BasicMatching | |
| namespace OAI | |
| /-! The integral Hadwiger bound from pairwise touching matchings. | |
| The small-branch counting argument is Proposition `matching-minor` of | |
| *A counterexample to Hadwiger's conjecture*, following the connected-matching | |
| approach of Cambie. | |
| -/ | |
| namespace HadwigerMatching | |
| variable {V : Type} [Fintype V] {G : SimpleGraph V} | |
| /-- The singleton and doubleton deficits of an actual clique-minor model are | |
| controlled by a touching matching constructed from those very branches. -/ | |
| theorem small_branch_deficit_le {t : ℕ} (B : Fin t → Finset V) | |
| (hconn : ∀ i, (G.induce (B i : Set V)).Connected) | |
| (hne : ∀ i, (B i).Nonempty) | |
| (hdisj : Pairwise (fun i j => Disjoint (B i) (B j))) | |
| (htouch : ∀ i j, i ≠ j → ∃ v ∈ B i, ∃ w ∈ B j, G.Adj v w) : | |
| 2 * singletonBranchCount B + doubletonBranchCount B ≤ | |
| 4 * connectedMatchingNumber G + 2 := by | |
| classical | |
| have hE := twoVertexBranches_isTouchingMatching G B hdisj htouch | |
| (fun i hi => connected_card_two_isClique (B i) hi (hconn i)) | |
| obtain ⟨P, hP, hcard, hcover⟩ := extend_matching_by_clique_pairs G | |
| (twoVertexBranches B) (singletonVertices B) hE | |
| (singletonVertices_isClique G B htouch) | |
| (fun _ he => twoVertexBranches_disjoint_singletonVertices B hdisj he) | |
| (fun _ hv _ he => singletonVertices_touch_twoVertexBranches G B htouch hv he) | |
| have hbound := matching_card_le_number hP | |
| rw [hcard, card_twoVertexBranches B hne hdisj] at hbound | |
| rw [card_singletonVertices B hdisj] at hcover | |
| omega | |
| /-- Every clique minor satisfies the integral matching bound. -/ | |
| theorem clique_minor_matching_bound {t : ℕ} | |
| (h : LinearListHadwiger.HasCliqueMinor G t) : | |
| 3 * t ≤ Fintype.card V + 4 * connectedMatchingNumber G + 2 := by | |
| classical | |
| obtain ⟨B, hconn, hne, hdisj, htouch⟩ := finite_branch_model h | |
| have hcount := branch_counting B hne hdisj | |
| have hsmall := small_branch_deficit_le B hconn hne hdisj htouch | |
| omega | |
| /-- The maximum is attained by an actual clique-minor branch model. This proves | |
| the bound for the literal Hadwiger number, including the empty graph. -/ | |
| theorem hadwiger_matching_bound (G : SimpleGraph V) : | |
| 3 * LinearListHadwiger.hadwigerNumber G ≤ | |
| Fintype.card V + 4 * connectedMatchingNumber G + 2 := | |
| clique_minor_matching_bound (LinearListHadwiger.hasCliqueMinor_hadwigerNumber G) | |
| end HadwigerMatching | |
| end OAI | |