Planar edge count upper bound

Proved

WilliamsonShmoys.planar_edge_count_upper_bound__p2m_0227f8b9_c5e1_4755_9e67_a93be5e3575c_a1

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

approximation-algorithmsplanar-graphstreewidth

Every graph with a crossing-free drawing and at least three vertices has at most 3|V|−6 edges. Prove this by relating vertices, edges, and faces through Euler's formula and using that each face boundary contributes at least three edge-sides (with the small or degenerate cases handled explicitly). This provides the upper inequality independently of maximality.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_edge_count_upper_bound__p2m_0227f8b9_c5e1_4755_9e67_a93be5e3575c_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (Q : SimpleGraph V)
    (hcard : 3 <= Fintype.card V)
    (hQ : WilliamsonShmoys.HasPlanarDrawingOn Q) :
    letI : DecidableRel Q.Adj := Classical.decRel Q.Adj
    Q.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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