Edge count of a maximal planar graph

Proved

WilliamsonShmoys.maximal_planar_graph_edge_count__p2m_76432751_4a60_438b_af25_c322d55d602e_a2

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

approximation-algorithmsplanar-graphstreewidth

A finite simple planar graph with at least three vertices that is inclusion-maximal among planar graphs on its vertex set has exactly $3|V|-6$ edges. A proof uses triangular faces, edge-face incidence counting, and Euler’s formula; the stated vertex bound includes the three-vertex boundary case.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_graph_edge_count__p2m_76432751_4a60_438b_af25_c322d55d602e_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hV : 3 ≤ Fintype.card V)
    (hmax : ∀ (K : SimpleGraph V) (dK : DecidableRel K.Adj),
      (letI : DecidableRel K.Adj := dK
       WilliamsonShmoys.HasPlanarDrawingOn K →
       H.edgeFinset ⊆ K.edgeFinset → K.edgeFinset ⊆ H.edgeFinset)) :
    H.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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