Planar drawing edge bound

Disproved

WilliamsonShmoys.planar_drawing_edge_bound__p2m_a6656023_52ae_443d_ab7b_e0c464fa9c80_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite simple graph with at least three vertices and a drawing satisfying HasPlanarDrawingOn has at least $3|V|-6$ edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_edge_bound__p2m_a6656023_52ae_443d_ab7b_e0c464fa9c80_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hM : HasPlanarDrawingOn M) :
    3 * Fintype.card V - 6 ≤
      (letI : DecidableRel M.Adj := Classical.decRel M.Adj
       M.edgeFinset.card) := 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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