Planar simple graph edge upper bound

Open

WilliamsonShmoys.planar_drawable_edge_upper_bound__p2m_7cf72e3d_7b9d_4e33_bfe4_614d5c224b98_a1

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

approximation-algorithmsplanar-graphstreewidth

Any finite simple graph on at least three vertices with a crossing-free drawing as defined by HasPlanarDrawingOn has at most 3n−6 edges. The order and decidable adjacency arguments are retained to match the target’s representation.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawable_edge_upper_bound__p2m_7cf72e3d_7b9d_4e33_bfe4_614d5c224b98_a1 {V : Type} [Fintype V] [LinearOrder V] (M : SimpleGraph V) (dM : DecidableRel M.Adj) (hM : HasPlanarDrawingOn M) (hcard : 3 <= Fintype.card V) : M.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).

View graph