Euler edge bound for a drawable planar graph

Proved

WilliamsonShmoys.planar_edge_bound__p2m_ac6eb2e4_2427_46c2_b7cd_156017af8bc0_a1

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

approximation-algorithmsplanar-graphstreewidth

A simple graph with a crossing-free drawing on at least three vertices has at most 3|V|−6 edges.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_edge_bound__p2m_ac6eb2e4_2427_46c2_b7cd_156017af8bc0_a1 {V : Type} [Fintype V] [LinearOrder V] (G : SimpleGraph V) (hDraw : WilliamsonShmoys.HasPlanarDrawingOn G) (hCard : 3 ≤ Fintype.card V) : G.edgeSet.ncard + 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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