Corrected Euler face certificate for a crossing-free drawing

Proved

WilliamsonShmoys.crossing_free_drawing_euler_face_certificate__p2m_81ed959b_71ee_4030_85d8_875841fa0344_a2

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For a finite connected simple graph with at least three vertices and a crossing-free drawing, some natural-number face count satisfies the planar Euler equation and the lower bound on total face-boundary length.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.crossing_free_drawing_euler_face_certificate__p2m_81ed959b_71ee_4030_85d8_875841fa0344_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hH : HasPlanarDrawingOn H)
    (hconn : H.Connected)
    (hcard : 3 <= Fintype.card V) :
    exists f : Nat,
      H.edgeSet.ncard + 2 = Fintype.card V + f /\
      3 * f <= 2 * H.edgeSet.ncard := 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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