Edge bound for a maximal planar drawing

Proved

WilliamsonShmoys.maximal_planar_edge_bound__p2m_1869d2b3_4acb_4df1_b926_664515403053_a2

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

approximation-algorithmsplanar-graphstreewidth

A maximal drawable simple graph on at least three vertices has at most 3n − 6 edges. The supplied definition’s arc conditions encode a crossing-free embedding. Maximality forces the embedding to be connected with triangular face boundaries; counting the two sides of each edge and applying Euler’s formula gives the bound. The n = 3 boundary is included and yields the bound for K₃. This removes the topological counting step from the parent; formalizing Euler’s formula and face-edge incidence for the arc-based drawing remains the core obligation.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_bound__p2m_1869d2b3_4acb_4df1_b926_664515403053_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hmax :
      ∀ K : SimpleGraph V,
        (∀ u v, H.Adj u v → K.Adj u v) →
        WilliamsonShmoys.HasPlanarDrawingOn K →
        ∀ u v, K.Adj u v → H.Adj u 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).

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