Edge bound for a drawable simple graph

Proved

WilliamsonShmoys.planar_simple_graph_edge_bound__p2m_89969936_4231_4118_a55c_9873f67c1fec_a1

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

approximation-algorithmsplanar-graphstreewidth

A simple graph with a crossing-free drawing on a finite linearly ordered vertex type has at most 3|V|−6 edges when |V|≥3. The condition includes the three-vertex boundary case.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_simple_graph_edge_bound__p2m_89969936_4231_4118_a55c_9873f67c1fec_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hdrawing : HasPlanarDrawingOn G) :
    letI : ∀ H : SimpleGraph V, DecidableRel H.Adj := fun H => Classical.decRel H.Adj
    G.edgeFinset.card ≤ 3 * Fintype.card V - 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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