Euler face count and incidence bound for a connected planar drawing

Proved

WilliamsonShmoys.connected_drawing_euler_face_count__p2m_94535f0f_7fcf_46ec_8860_fc174f378d1a_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 specified crossing-free drawing, there is a natural number of faces satisfying Euler’s formula and the lower bound on total face-boundary incidences.

Preamble

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

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.connected_drawing_euler_face_count__p2m_94535f0f_7fcf_46ec_8860_fc174f378d1a_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hcard : 3 <= Fintype.card V)
    (hconn : G.Connected)
    (hG : WilliamsonShmoys.HasPlanarDrawingOn G) :
    ∃ f : ℕ,
      f + Fintype.card V = 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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