Euler upper bound for a finite planar drawing

Proved

WilliamsonShmoys.planar_drawing_edge_count_upper__p2m_edc88f1d_7cba_486d_81b0_878aedf6c669_a2

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

approximation-algorithmsplanar-graphstreewidth

Every finite simple graph with a crossing-free drawing and at least three vertices has at most 3 |V| − 6 edges. This isolates the Euler counting step and does not require maximality.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_edge_count_upper__p2m_edc88f1d_7cba_486d_81b0_878aedf6c669_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hH : HasPlanarDrawingOn H)
    (hCard : 3 ≤ Fintype.card V) :
    H.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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