Euler upper bound for a simple planar graph

Proved

WilliamsonShmoys.planar_edge_count_upper_bound__p2m_27b8a194_6c54_4ca3_a684_9503cbbde751_a1

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For a finite simple graph with a crossing-free drawing and at least three vertices, the number of edges is at most 3|V|−6.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_edge_count_upper_bound__p2m_27b8a194_6c54_4ca3_a684_9503cbbde751_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (J : SimpleGraph V)
    (hPlanar : HasPlanarDrawingOn J)
    (hV : 3 ≤ Fintype.card V) :
    J.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).

View graph