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Parallel cluster splitting

Lax17.ParallelClusterSplitting · concepts/Lax17/ParallelClusterSplitting.lean · lax-17

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    Natural Language Statement

    Theorem

    Three large linked terminal sets in one cluster can be retained in three pairwise disjoint connected subclusters.

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

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax17.PathOfSets
    2import Lax17.Degree
    3
    4/-!
    5---
    6title: Parallel cluster splitting
    7type: theorem
    8---
    9Three large linked terminal sets in one cluster can be retained in three
    10pairwise disjoint connected subclusters.
    11-/
    12
    13namespace Lax17.ParallelClusterSplitting
    14
    15universe u
    16
    17/-- The degree-three cluster-splitting theorem used to create one base/hair
    18pair while retaining linked interfaces of width `w`. -/
    19axiom parallelClusterSplitting :
    20 ∃ c : ℕ, 0 < c ∧
    21 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    22 (G : SimpleGraph V) {C A B : Finset V} {w : ℕ},
    23 0 < w →
    24 Lax17.Degree.MaximumAtMost G 3 →
    25 Lax17.PathOfSets.IsCluster G C →
    26 A ⊆ C →
    27 B ⊆ C →
    28 A.card = c * w →
    29 B.card = c * w →
    30 Disjoint A B →
    31 Lax17.Linkedness.NodeWellLinkedIn G C A →
    32 Lax17.Linkedness.NodeWellLinkedIn G C B →
    33 Lax17.Linkedness.NodeLinkedIn G C A B →
    34 Nonempty
    35 (Lax17.PathOfSets.HairyClusterSplit
    36 G C A B w)
    37
    38end Lax17.ParallelClusterSplitting
    39
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