Edge count of a maximal drawable graph

Proved

WilliamsonShmoys.maximal_planar_exact_edge_count__p2m_bba3ebb6_f48a_40f6_8fc3_19f0d2d80e0e_a1

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

approximation-algorithmsplanar-graphstreewidth

An edge-maximal drawable simple graph on n ≥ 3 vertices has exactly 3n − 6 edges. Maximality rules out missing edges that could be drawn inside faces, so the embedding is a triangulation; Euler’s formula then gives the stated count. For n = 3, the only maximal case is the triangle, giving the boundary value 3.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_exact_edge_count__p2m_bba3ebb6_f48a_40f6_8fc3_19f0d2d80e0e_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hK : HasPlanarDrawingOn K) :
    letI : ∀ L : SimpleGraph V, DecidableRel L.Adj :=
      fun L => Classical.decRel L.Adj
    (∀ L : SimpleGraph V,
      K.edgeFinset ⊆ L.edgeFinset →
      HasPlanarDrawingOn L →
      L.edgeFinset = K.edgeFinset) →
    K.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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