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Planar graphs have a vertex of degree at most 5

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

proven

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    Natural Language Statement

    Theorem

    Every finite planar graph with at least one vertex has a vertex of degree at most 55.

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

    Each proof establishes this claim relative to its assumptions.

    In the paper

    • page 2 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 a vertex of degree at most 5
    7type: theorem
    8---
    9Every finite planar graph with at least one vertex has a vertex of degree at
    10most 55.
    11-/
    12
    13set_option autoImplicit false
    14
    15namespace Lax909950.LowDegree
    16
    17/-- Every nonempty finite planar graph has a vertex with at most five
    18neighbors. -/
    19axiom exists_degree_le_five {V : Type*} [Finite V] [Nonempty V] {G : SimpleGraph V}
    20 (hG : Lax68.Planar.IsPlanar G) :
    21 ∃ x : V, (G.neighborSet x).ncard ≤ 5
    22
    23end Lax909950.LowDegree
    24
    Show Proof
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