Maximality forces the planar edge lower bound

Proved

WilliamsonShmoys_maximal_planar_graph_edge_count_lower_bound__p2m_f302bffe_a312_4c80_ba34_5d3904b8242e_a1

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

approximation-algorithmsplanar-graphstreewidth

An inclusion-maximal finite simple planar graph on at least three vertices has at least 3n−6 edges. Maximality excludes adding any edge while preserving a planar drawing; the standard route is to show the drawing is connected and every face is triangular, then use Euler's formula and edge-face incidence. This supplies the lower-counting half independently of the Euler upper bound.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys_maximal_planar_graph_edge_count_lower_bound__p2m_f302bffe_a312_4c80_ba34_5d3904b8242e_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hL : WilliamsonShmoys.HasPlanarDrawingOn L)
    (hMax : ∀ M : SimpleGraph V,
      L ≤ M → WilliamsonShmoys.HasPlanarDrawingOn M → M = L) :
    3 * Fintype.card V - 6 ≤ L.edgeSet.ncard := 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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