Edge bound for maximal planar graphs

Proved

WilliamsonShmoys.maximal_planar_edge_bound__p2m_b05b2ac9_cf86_42cd_81eb_d15df4254060_a2

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

approximation-algorithmsplanar-graphstreewidth

A maximal planar simple graph on a finite vertex type with at least three vertices has at most 3n − 6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_bound__p2m_b05b2ac9_cf86_42cd_81eb_d15df4254060_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hK : HasPlanarDrawingOn K)
    (hMaximal : ∀ L : SimpleGraph V, K ≤ L → HasPlanarDrawingOn L → L = K) :
    (letI : DecidableRel K.Adj := Classical.decRel K.Adj;
      K.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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