Planar graphs satisfy the maximum edge bound

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WilliamsonShmoys.planar_edge_card_le_max__p2m_ba4159d3_5323_4426_8c82_d674db3cc476_a1

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

approximation-algorithmsplanar-graphstreewidth

A graph with a crossing-free drawing on a finite ordered vertex set has at most the complete-graph edge count when there are at most two vertices, and at most 3|V|−6 edges otherwise.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_edge_card_le_max__p2m_ba4159d3_5323_4426_8c82_d674db3cc476_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (dG : DecidableRel G.Adj)
    (hG : HasPlanarDrawingOn G) :
    G.edgeFinset.card ≤
      if Fintype.card V ≤ 2 then
        Fintype.card V * (Fintype.card V - 1) / 2
      else 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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