Euler face count and facial degree bound from a planar drawing

Proved

WilliamsonShmoys.planar_face_count_from_drawing__p2m_c508f2ea_079b_4849_b4cc_0b3b748b75aa_a1

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

approximation-algorithmsplanar-graphstreewidth

For a connected simple graph on at least three vertices with the given crossing-free drawing, there is a finite face count satisfying Euler’s formula and the bound that total facial degree is at least three per face.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_face_count_from_drawing__p2m_c508f2ea_079b_4849_b4cc_0b3b748b75aa_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (dH : DecidableRel H.Adj)
    (hconn : H.Connected)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hn : Nat.le 3 (Fintype.card V)) :
    ∃ f : Nat,
      H.edgeFinset.card + 2 = Fintype.card V + f ∧
      3 * f ≤ 2 * H.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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