Edge lower bound for a maximal drawable graph

Proved

WilliamsonShmoys.maximal_planar_edge_count__p2m_9cc0c9d9_a289_45c0_90c4_3b812fbd190b_a1

by Eyal1990 · Oct 29, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

An inclusion-maximal drawable simple graph on at least three vertices has at least 3|V|−6 edges.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_9cc0c9d9_a289_45c0_90c4_3b812fbd190b_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V)
    (hV : 3 ≤ Fintype.card V)
    (hM : WilliamsonShmoys.HasPlanarDrawingOn M)
    (hmax : ∀ N : SimpleGraph V,
      Set.Subset M.edgeSet N.edgeSet →
      WilliamsonShmoys.HasPlanarDrawingOn N → N = M) :
    3 * Fintype.card V ≤ M.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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