Planar drawing edge upper bound

Proved

WilliamsonShmoys.edge_count_le_of_planar_drawing

by Eyal1990 · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Every graph with the supplied crossing-free drawing on at least three vertices has at most 3n − 6 edges. The boundary n = 3 is included; for disconnected drawings the proof must account for all components rather than assume connectedness. This removes the universal planar edge upper-bound calculation from the parent. A proof route is to derive Euler’s formula for the drawing, use that each face boundary has length at least three after handling tree components and bridges, and count face-edge incidences; alternatively, augment components to a connected planar graph before applying the standard bound. This is a new formal argument for the given drawing predicate; no matching Lean declaration was available locally.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.edge_count_le_of_planar_drawing
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hV : 3 ≤ Fintype.card V)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn G) :
    G.edgeSet.ncard ≤ 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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