Edge count of a maximal planar graph

Proved

WilliamsonShmoys.maximal_planar_edge_count__p2m_54b85171_3c84_49d0_b75e_a1ac45808ba4_a2

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

approximation-algorithmsplanar-graphstreewidth

An inclusion-maximal planar simple graph on at least three vertices has 3|V|−6 edges. This supplies the counting step in the parent. A proof route is to establish connectedness and triangular faces from maximality, then use the planar Euler formula and count edge-face incidences; the main remaining work is formalizing those embedding and face-count arguments for the given crossing-free arc predicate.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_54b85171_3c84_49d0_b75e_a1ac45808ba4_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (hplanar : WilliamsonShmoys.HasPlanarDrawingOn N)
    (hmax : ∀ K : SimpleGraph V,
      WilliamsonShmoys.HasPlanarDrawingOn K → N ≤ K → K = N)
    (hcard : 3 ≤ Fintype.card V) :
    N.edgeSet.ncard + 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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