An edge-maximal planar graph has the Euler bound with equality

Proved

WilliamsonShmoys.maximal_planar_edge_count__p2m_711cef84_2190_482b_bc2d_a11a812cbece_a1

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

approximation-algorithmsplanar-graphstreewidth

A planar graph on at least three vertices that is maximal among planar supergraphs on the same vertex type has exactly three times the number of vertices minus six edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_711cef84_2190_482b_bc2d_a11a812cbece_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hH : HasPlanarDrawingOn H)
    (hmax : ∀ K : SimpleGraph V, HasPlanarDrawingOn K →
      (∀ u v, H.Adj u v → K.Adj u v) →
      ∀ u v, K.Adj u v → H.Adj u v) :
    (letI : DecidableRel H.Adj := Classical.decRel H.Adj;
      H.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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