Upper edge bound for simple planar graphs

Proved

WilliamsonShmoys.planar_edge_count_upper_bound__p2m_4d9bb183_475b_4a59_8b75_1a8cda8dae1d_a2

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A finite simple graph on at least three vertices with a crossing-free drawing has at most 3|V|−6 edges, by the planar Euler edge bound.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_edge_count_upper_bound__p2m_4d9bb183_475b_4a59_8b75_1a8cda8dae1d_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (dL : DecidableRel L.Adj)
    (hL : HasPlanarDrawingOn L)
    (hcard : 3 <= Fintype.card V) :
    letI : DecidableRel L.Adj := dL
    L.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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