Edge count of a maximal planar graph

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WilliamsonShmoys.maximal_planar_edge_count__p2m_2a77e1c0_132e_4864_a8ba_ceda5e75c647_a2

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

approximation-algorithmsplanar-graphstreewidth

A finite connected inclusion-maximal simple planar graph with at least three vertices has exactly 3n − 6 edges. A proof route is to show maximality makes every face triangular, then combine the face-degree sum with Euler’s formula.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_2a77e1c0_132e_4864_a8ba_ceda5e75c647_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hplanar : HasPlanarDrawingOn H)
    (hconn : H.Connected)
    (hcard : 3 <= Fintype.card V)
    (hmax : forall K : SimpleGraph V, H <= K -> HasPlanarDrawingOn K -> K = H) :
    H.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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