Simple planar edge upper bound

Proved

WilliamsonShmoys.planar_edge_upper_bound__p2m_d7b584f5_f48a_4e68_a764_d92a7993a6d3_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite simple graph with at least three vertices has at most three times its vertex count minus six edges. This is the standard planar edge upper bound, obtained from Euler’s formula and the fact that each face boundary has length at least three, with the usual treatment of trees and bridges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_edge_upper_bound__p2m_d7b584f5_f48a_4e68_a764_d92a7993a6d3_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hL : HasPlanarDrawingOn L) :
    letI : DecidableRel L.Adj := Classical.decRel L.Adj
    L.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).

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