Finite face count from a crossing-free graph drawing

Proved

WilliamsonShmoys.planar_drawing_face_count__p2m_45b42f5d_e929_44e5_b683_b600a07db038_a3

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

approximation-algorithmsplanar-graphstreewidth

A connected simple graph with at least three vertices and the supplied crossing-free drawing has a finite face count satisfying Euler’s equation and the standard lower bound on total face degree.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_face_count__p2m_45b42f5d_e929_44e5_b683_b600a07db038_a3 {V : Type} [Fintype V] [LinearOrder V] (M : SimpleGraph V) [DecidableRel M.Adj] (hM : WilliamsonShmoys.HasPlanarDrawingOn M) (hconn : M.Connected) (hcard : 3 ≤ Fintype.card V) : ∃ f : ℕ, Fintype.card V + f = M.edgeFinset.card + 2 ∧ 3 * f ≤ 2 * M.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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