Euler face count for a connected planar drawing

Proved

WilliamsonShmoys.connected_planar_face_count__p2m_69e73f60_1756_4e30_afbd_efe3ac2bd3ed_a1

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

approximation-algorithmsplanar-graphstreewidth

A connected simple graph with at least three vertices and a crossing-free drawing has a natural-number face count satisfying Euler’s formula and the lower bound of three edge incidences per face.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.connected_planar_face_count__p2m_69e73f60_1756_4e30_afbd_efe3ac2bd3ed_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hN : WilliamsonShmoys.HasPlanarDrawingOn N)
    (hconn : N.Connected) :
    (letI : DecidableRel N.Adj := Classical.decRel N.Adj
     ∃ f : ℕ,
       Fintype.card V + f = N.edgeFinset.card + 2 ∧
       3 * f ≤ 2 * N.edgeFinset.card) := 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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