Planar graph edge-count upper bound

Proved

WilliamsonShmoys.maximal_planar_supergraph_edge_count_upper__p2m_126bf562_f451_4e0b_b87e_ef5c21746e7b_a1

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

approximation-algorithmsplanar-graphstreewidth

Any graph with a crossing-free drawing and at least three vertices has at most 3|V|−6 edges. This isolates the Euler-formula and face-incidence argument needed for the upper inequality.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_supergraph_edge_count_upper__p2m_126bf562_f451_4e0b_b87e_ef5c21746e7b_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hP : HasPlanarDrawingOn P) :
    (letI : DecidableRel P.Adj := Classical.decRel P.Adj;
      P.edgeFinset.card + 6 ≤ 3 * Fintype.card V) := 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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