Euler’s upper edge bound for a drawable graph

Proved

WilliamsonShmoys.planar_edge_card_upper_bound__p2m_078d72d1_c4b8_4d42_8ead_9d2688891836_a1

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A simple graph with a crossing-free drawing on at least three finite ordered vertices has at most three times the number of vertices minus six edges.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_edge_card_upper_bound__p2m_078d72d1_c4b8_4d42_8ead_9d2688891836_a1 {V : Type} [Fintype V] [LinearOrder V] (K : SimpleGraph V) [DecidableRel K.Adj] (hcard : 3 <= Fintype.card V) (hK : WilliamsonShmoys.HasPlanarDrawingOn K) : K.edgeFinset.card <= 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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