Euler and face-length certificate for a connected planar drawing

Proved

WilliamsonShmoys.planar_drawing_euler_face_certificate__p2m_3061538d_e777_4f56_997f_0c12065c9e75_a1

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

approximation-algorithmsplanar-graphstreewidth

For a connected simple graph with at least three vertices and a crossing-free drawing, there is a natural number of faces satisfying Euler’s formula and the lower bound of three on the total face-boundary length.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_euler_face_certificate__p2m_3061538d_e777_4f56_997f_0c12065c9e75_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hplanar : HasPlanarDrawingOn G)
    (hconn : G.Connected)
    (hcard : 3 <= Fintype.card V) :
    ∃ f : ℕ,
      Fintype.card V + f = G.edgeSet.ncard + 2 ∧
      3 * f <= 2 * G.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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