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Balanced closed parts certify a balanced separator

Lax263307.BalancedPartitions · concepts/Lax263307/BalancedPartitions.lean · lax-263307

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    Natural Language Statement

    Theorem

    A partition of undeleted vertices into parts closed under undeleted graph edges places each connected component wholly inside one part. If every part has at most two thirds of the original vertices, the deleted set is a balanced separator. This permits the parts supplied by repeated cuts.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    Lean source view on GitHub

    1import Lax263307.SeparatorComponents
    2import Lax263307.EdgeSublinearSeparators
    3
    4/-!
    5---
    6title: Balanced closed parts certify a balanced separator
    7type: theorem
    8---
    9A partition of undeleted vertices into parts closed under undeleted graph
    10edges places each connected component wholly inside one part. If every
    11part has at most two thirds of the original vertices, the deleted set is
    12a balanced separator. This permits the parts supplied by repeated cuts.
    13
    14# Attribution
    15
    16The balanced-component conclusion of Bonnet–Chang's repeated-cut Lemma 8.
    17-/
    18
    19namespace Lax263307.BalancedPartitions
    20
    21open Lax263307.SeparatorComponents Lax263307.EdgeSublinearSeparators
    22
    23axiom balanced_separator {n : ℕ} (G : SimpleGraph (Fin n)) (S : Finset (Fin n))
    24 {ι : Type} (P : Partition G S ι) (balanced : ∀ i, 3 * (P.vertices i).card ≤ 2 * n) :
    25 IsBalancedSeparator G S
    26
    27end Lax263307.BalancedPartitions
    28
    Show Proof

    Attribution

    The balanced-component conclusion of Bonnet–Chang's repeated-cut Lemma 8.

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