Planar drawing edge upper bound

Proved

WilliamsonShmoys.maximal_planar_drawing_edge_count_upper__p2m_2fe61b07_5675_43dd_a849_e69c0c6246fb_a1

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

approximation-algorithmsplanar-graphstreewidth

Every simple graph with the specified crossing-free drawing and at least three vertices has at most 3|V| - 6 edges. This child removes the universal upper-bound part of the parent count; it follows by Euler’s formula and face-degree counting, with disconnected components handled separately or connected by added arcs.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_edge_count_upper__p2m_2fe61b07_5675_43dd_a849_e69c0c6246fb_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (hV : 3 <= Fintype.card V)
    (hN : HasPlanarDrawingOn N) :
    letI : DecidableRel N.Adj := Classical.decRel N.Adj
    N.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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