Simple planar graph edge bound

Proved

WilliamsonShmoys.planar_drawing_edge_bound__p2m_b3c9bfa4_c900_4902_b62c_e651fcb234b7_a1

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

approximation-algorithmsplanar-graphstreewidth

A connected simple graph with at least three vertices and a crossing-free drawing has at most three times its vertex count minus six edges. The additive form avoids truncated subtraction in the natural-number statement.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_edge_bound__p2m_b3c9bfa4_c900_4902_b62c_e651fcb234b7_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hcard : 3 <= Fintype.card V)
    (hconn : G.Connected)
    (hG : WilliamsonShmoys.HasPlanarDrawingOn G) :
    G.edgeSet.ncard + 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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