Lax Lean Archive

Arbitrarily Large Subcubic Graphs of Linear Treewidth

Lax920634.TreewidthExistence · concepts/Lax920634/TreewidthExistence.lean · lax-920634

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Theorem

    For every size threshold there is a finite simple subcubic graph above that threshold with treewidth at least its order divided by 48d+248d+2, for the same fixed degree constant d=D2d=D^2. These graphs belong to the explicit expander family.

    Concept map
    6 concepts
    100%
    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 Lax920634.ZigZagConstruction
    2import Lax228581.Treewidth
    3
    4/-!
    5---
    6title: Arbitrarily Large Subcubic Graphs of Linear Treewidth
    7type: theorem
    8---
    9For every size threshold there is a finite simple subcubic graph above that
    10threshold with treewidth at least its order divided by 48d+248d+2, for the same
    11fixed degree constant d=D2d=D^2. These graphs belong to the explicit expander
    12family.
    13
    14# Formalization notes
    15
    16The integer inequality n≤(48d+2)tw⁡(G)n\le(48d+2)\operatorname{tw}(G) expresses the
    17linear lower bound without division or rounding. The proof selects a large
    18enough graph to absorb the additive one in the general treewidth inequality.
    19-/
    20
    21namespace Lax920634.TreewidthExistence
    22
    23open ZigZagConstruction EdgeExpansion Lax228581.Treewidth
    24
    25/-- Arbitrarily large subcubic graphs have a fixed linear lower bound on treewidth. -/
    26axiom exists_subcubic_linear_treewidth (lower : ℕ) :
    27 ∃ n : ℕ, ∃ G : SimpleGraph (Fin n), lower ≤ n ∧ IsSubcubic G ∧
    28 n ≤ (48 * degree + 2) * treewidth G
    29
    30end Lax920634.TreewidthExistence
    31
    Show Proof
    Formalization notes

    The integer inequality n≤(48d+2)tw⁡(G)n\le(48d+2)\operatorname{tw}(G) expresses the linear lower bound without division or rounding. The proof selects a large enough graph to absorb the additive one in the general treewidth inequality.

    Builds on
    Used by

    none

    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…