Edge lower bound for an edge-maximal drawable graph

Proved

WilliamsonShmoys.maximal_drawable_edge_lower_bound__p2m_84b83a52_4380_4290_8494_df49bd55089b_a1

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

approximation-algorithmsplanar-graphstreewidth

An edge-maximal finite drawable simple graph on at least three vertices has at least 3n−6 edges; maximality rules out nontriangular faces in a planar embedding.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_drawable_edge_lower_bound__p2m_84b83a52_4380_4290_8494_df49bd55089b_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hN : WilliamsonShmoys.HasPlanarDrawingOn N)
    (hmax : ∀ P : SimpleGraph V,
      (∀ u v, N.Adj u v → P.Adj u v) →
      WilliamsonShmoys.HasPlanarDrawingOn P →
      ∀ u v, P.Adj u v → N.Adj u v) :
    3 * Fintype.card V ≤ N.edgeSet.ncard + 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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