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Degree-three treewidth sparsifier

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

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    Natural Language Statement

    Theorem

    Large treewidth contains a degree-three spanning subgraph that preserves treewidth up to a polylogarithmic factor.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    1import Mathlib.Data.Nat.Log
    2import Lax17.Degree
    3import Lax17.Treewidth
    4
    5/-!
    6---
    7title: Degree-three treewidth sparsifier
    8type: theorem
    9---
    10Large treewidth contains a degree-three spanning subgraph that preserves
    11treewidth up to a polylogarithmic factor.
    12-/
    13
    14namespace Lax17.TreewidthSparsifier
    15
    16universe u
    17
    18/-- If `G` has treewidth at least `k > 1`, it has a spanning subgraph `H` of
    19maximum degree three for which
    20`k ≤ c · treewidth(H) · (log₂ k)^d`. -/
    21axiom degreeThreeTreewidthSparsifier :
    22 ∃ c d : ℕ, 0 < c ∧ 0 < d ∧
    23 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    24 (G : SimpleGraph V) {k : ℕ},
    25 1 < k →
    26 k ≤ Lax17.Treewidth.treewidth G →
    27 ∃ H : SimpleGraph V,
    28 H ≤ G ∧
    29 Lax17.Degree.MaximumAtMost H 3 ∧
    30 k ≤ c * Lax17.Treewidth.treewidth H *
    31 (Nat.log 2 k) ^ d
    32
    33end Lax17.TreewidthSparsifier
    34
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