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Lax17.EdgeMenger

Edge-Menger theorem

concepts/Lax17/EdgeMenger.lean · lax-17

proven

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    Theorem

    The edge form of Menger's theorem gives an exact alternative between kk edge-disjoint paths inside a cluster and a cut partition with boundary of size less than kk.

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    1import Lax17.Paths
    2
    3/-!
    4---
    5title: Edge-Menger theorem
    6type: theorem
    7---
    8The edge form of Menger's theorem gives an exact alternative between
    9\(k\) edge-disjoint paths inside a cluster and a cut partition with boundary
    10of size less than \(k\).
    11-/
    12
    13namespace Lax17.EdgeMenger
    14
    15universe u
    16
    17/-- Finite edge-Menger in packing-or-cut form. -/
    18axiom edgeMenger :
    19 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    20 (G : SimpleGraph V) (C A B : Finset V) (k : ℕ),
    21 A ⊆ C →
    22 B ⊆ C →
    23 Disjoint A B →
    24 (∃ P : Lax17.Paths.EdgeLinkage G A B k,
    25 ∀ i : Fin k, (P.path i).StaysIn C) ∨
    26 Nonempty (Lax17.Paths.EdgeCutPartition G C A B k)
    27
    28end Lax17.EdgeMenger
    29
    Show Proof

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