Simple planar graph edge upper bound

Open

WilliamsonShmoys.planarEdgeUpperBound__p2m_22365758_ec9b_4f13_a59a_216cdd6c8044_a1__p2m_22365758_ec9b_4f13_a59a_216cdd6c8044_a5

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A finite simple graph with a crossing-free drawing and at least three vertices has at most 3|V| − 6 edges. This removes the numerical upper-bound argument from the parent; a proof route uses Euler’s formula and face counting, treating disconnected drawings as well.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planarEdgeUpperBound__p2m_22365758_ec9b_4f13_a59a_216cdd6c8044_a1__p2m_22365758_ec9b_4f13_a59a_216cdd6c8044_a5
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (dG : DecidableRel G.Adj)
    (hG : HasPlanarDrawingOn G)
    (hn : 3 <= Fintype.card V) :
    letI : DecidableRel G.Adj := dG
    G.edgeFinset.card <= 3 * Fintype.card V - 6 := by sorry

Source

David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).

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