Euler upper bound for finite simple planar graphs

Proved

WilliamsonShmoys.planar_simple_graph_edge_upper_bound__p2m_438c2037_6b64_401b_a3fa_84e1ccf42d6c_a1

by Eyal1990 · Nov 1, 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 is the generic planar upper bound, proved by connecting components as needed and applying Euler's formula with the planar face-boundary count, handling low-edge cases separately.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

Formal statement

theorem WilliamsonShmoys.planar_simple_graph_edge_upper_bound__p2m_438c2037_6b64_401b_a3fa_84e1ccf42d6c_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
    (hcard : 3 ≤ Fintype.card V) :
    G.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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