Planar edge upper bound

Proved

WilliamsonShmoys.maximal_planar_edge_count_upper__p2m_32302080_d9f8_461c_9542_c9cd7d932e42_a1

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

approximation-algorithmsplanar-graphstreewidth

For a graph with the target’s drawing and at least three vertices, its edge count is at most three times the number of vertices minus six. Maximality is retained in the statement for direct compatibility, though the Euler bound itself does not need it.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count_upper__p2m_32302080_d9f8_461c_9542_c9cd7d932e42_a1 {V : Type} [Fintype V] [LinearOrder V] (R : SimpleGraph V) (hR : HasPlanarDrawingOn R) (hcard : 3 ≤ Fintype.card V) (hmax : ∀ S : SimpleGraph V, R ≤ S → HasPlanarDrawingOn S → S = R) : ∀ [hRdec : DecidableRel R.Adj], R.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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