Planar edge-count upper bound

Proved

WilliamsonShmoys.planar_edge_count_upper_bound__p2m_f84a2b8d_758d_48e3_856e_f835c2dfd776_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable simple graph on at least three vertices has at most three times its vertex count minus six edges. This supplies the upper inequality. A proof route uses Euler’s formula and face-boundary counting, with a separate reduction or count for disconnected graphs.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_edge_count_upper_bound__p2m_f84a2b8d_758d_48e3_856e_f835c2dfd776_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 + 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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