Maximal planar graphs have the triangulation edge count

Proved

WilliamsonShmoys.maximal_planar_edge_count__p2m_2f35ea45_99df_4cf1_95b2_81f89b32ddea_a1

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

approximation-algorithmsplanar-graphstreewidth

A finite graph with at least three vertices that is maximal among planar supergraphs on the same vertex type has exactly 3|V|−6 edges. The proof route is to show maximality forces every face of a planar drawing to be triangular, then combine 3|F|=2|E| with Euler’s formula |V|−|E|+|F|=2.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_2f35ea45_99df_4cf1_95b2_81f89b32ddea_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hK : HasPlanarDrawingOn K)
    (hmax : ∀ L : SimpleGraph V, K ≤ L → HasPlanarDrawingOn L → L = K)
    (hn : 3 <= Fintype.card V) :
    letI : DecidableRel K.Adj := Classical.decRel _
    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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