Face-incidence bound for a simple planar drawing

Proved

WilliamsonShmoys.planar_drawing_face_length_bound__p2m_abda76d0_f206_4b5c_973f_e20fb605b671_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, any natural face count satisfying Euler's equation obeys the face-incidence inequality. A proof route is to represent each face by its boundary walk, show each such walk has length at least three, and count two incidences per edge.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_face_length_bound__p2m_abda76d0_f206_4b5c_973f_e20fb605b671_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hplanar : HasPlanarDrawingOn G)
    (hconn : G.Connected)
    (hcard : 3 <= Fintype.card V)
    (f : ℕ)
    (hEuler : 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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