Euler upper bound for a planar simple graph

Proved

WilliamsonShmoys.maximal_planar_upper_bound__p2m_8514b45e_f592_4fc5_88fb_d0f4ef7fe022_a1

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Every simply planar graph on at least three vertices has at most 3|V|−6 edges. Prove this from Euler’s formula and face-boundary counting; maximality is not needed.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_upper_bound__p2m_8514b45e_f592_4fc5_88fb_d0f4ef7fe022_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hcard : 3 ≤ Fintype.card V) :
    letI : DecidableRel H.Adj := Classical.decRel H.Adj
    H.edgeFinset.card + 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).

View graph