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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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | 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
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