Euler edge bound for planar drawings

Proved

WilliamsonShmoys.planar_drawing_edge_bound__p2m_3b70d152_dbfb_4f09_8d03_767be9da3cd3_a2

by Eyal1990 · Oct 31, 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 three times its number of vertices minus six edges, whether or not it is connected. Prove this by applying Euler’s formula componentwise, with separate bounds for components of size one or two.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_edge_bound__p2m_3b70d152_dbfb_4f09_8d03_767be9da3cd3_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hcard : 3 <= Fintype.card V)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H) :
    H.edgeSet.ncard <= 3 * Fintype.card V - 6 := 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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