Planar drawing edge bound

Proved

WilliamsonShmoys.planar_drawing_edge_bound__p2m_88a3bf29_1582_4605_aca9_2743ecb4531b_a1

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

approximation-algorithmsplanar-graphstreewidth

Every graph admitting the supplied crossing-free drawing, on at least three vertices, has at most three times its number of vertices minus six edges. The proof route is Euler’s formula for planar embeddings, accounting for disconnected graphs and faces with short boundary walks. This removes the upper-bound half of the target; its core obligation is to formalize the planar Euler bound for this drawing predicate.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_edge_bound__p2m_88a3bf29_1582_4605_aca9_2743ecb4531b_a1 {V : Type} [Fintype V] [LinearOrder V] (G : SimpleGraph V) (hDraw : WilliamsonShmoys.HasPlanarDrawingOn G) (hV : 3 ≤ Fintype.card V) : G.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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