Maximal planar edge lower bound

Proved

WilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_2c52a5e8_a177_4eaa_adc1_eeae3ac4d632_a2

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

approximation-algorithmsplanar-graphstreewidth

A finite edge-maximal simple planar graph with at least three vertices has at least 3n−6 edges. Together with the planar upper bound, this forces equality.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_2c52a5e8_a177_4eaa_adc1_eeae3ac4d632_a2 {V : Type} [Fintype V] [LinearOrder V] (M : SimpleGraph V) (hplanar : WilliamsonShmoys.HasPlanarDrawingOn M) (hcard : 3 <= Fintype.card V) (hmax : forall M' : SimpleGraph V, M <= M' -> WilliamsonShmoys.HasPlanarDrawingOn M' -> M' = M) : 3 * Fintype.card V <= M.edgeSet.ncard + 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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