Simple planar graph edge bound

Proved

WilliamsonShmoys.planarDrawing_edge_bound__p2m_7e15a8d6_7041_4a6f_b660_410e45ab60c6_a1

by Eyal1990 · Oct 31, 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 3n − 6 edges, expressed without natural subtraction as m + 6 ≤ 3n. The size hypothesis is necessary for this bound.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planarDrawing_edge_bound__p2m_7e15a8d6_7041_4a6f_b660_410e45ab60c6_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (dG : DecidableRel G.Adj)
    (hG : HasPlanarDrawingOn G)
    (hcard : 3 ≤ Fintype.card V) :
    (letI : DecidableRel G.Adj := dG; G.edgeFinset.card) + 6 ≤ 3 * Fintype.card V := 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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