Planar edge bound for at least three vertices

Proved

WilliamsonShmoys.planar_drawing_edge_bound_of_card_ge_three__p2m_511609d8_7f6a_4ba7_bd36_d7671338159d_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite simple graph with a crossing-free drawing and at least three vertices, the number of edges is at most three times the number of vertices minus six. This isolates the Euler-formula argument that carries the main difficulty of the target.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_edge_bound_of_card_ge_three__p2m_511609d8_7f6a_4ba7_bd36_d7671338159d_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (dL : DecidableRel L.Adj)
    (hL : WilliamsonShmoys.HasPlanarDrawingOn L)
    (hcard : 3 ≤ Fintype.card V) :
    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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