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Planar graphs have at most 3v − 6 edges

Lax909950.EdgeDensity · concepts/Lax909950/EdgeDensity.lean · lax-909950

proven

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    Natural Language Statement

    Theorem

    Every finite planar graph with v≥3v \geq 3 vertices has at most 3v−63v - 6 edges.

    Concept map
    4 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    In the paper

    • page 1 of this submission's paper

    Lean source view on GitHub

    1import Mathlib.Data.Set.Card
    2import Lax68.Planar
    3
    4/-!
    5---
    6title: Planar graphs have at most 3v − 6 edges
    7type: theorem
    8---
    9Every finite planar graph with v≥3v \geq 3 vertices has at most 3v−63v - 6 edges.
    10-/
    11
    12set_option autoImplicit false
    13
    14namespace Lax909950.EdgeDensity
    15
    16/-- A finite planar graph with v≥3v \geq 3 vertices has at most 3v−63v - 6 edges.
    17(Since v≥3v \geq 3, the natural-number subtraction does not truncate.) -/
    18axiom edge_density {V : Type*} [Finite V] {G : SimpleGraph V}
    19 (hG : Lax68.Planar.IsPlanar G) (hV : 3 ≤ Nat.card V) :
    20 G.edgeSet.ncard ≤ 3 * Nat.card V - 6
    21
    22end Lax909950.EdgeDensity
    23
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