A planar drawing yields Euler and face-incidence counts

Proved

WilliamsonShmoys.planarDrawingOn_eulerIncidence__p2m_5781120e_0dd2_40db_ad87_76e7c8291b92_a1

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

approximation-algorithmsplanar-graphstreewidth

A crossing-free drawing of a finite connected simple graph with at least three vertices yields a number of faces and total face-boundary incidences satisfying Euler’s equation, two incidences per edge, and at least three incidences per face. A proof route derives a rotation system and its face walks from the drawing, applies Euler’s formula, and handles bridges by counting both sides. This removes the topological step from the parent; constructing and verifying those data is the remaining core obligation.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planarDrawingOn_eulerIncidence__p2m_5781120e_0dd2_40db_ad87_76e7c8291b92_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (dL : DecidableRel L.Adj)
    (hL : HasPlanarDrawingOn L)
    (hconn : L.Connected)
    (hn : 3 ≤ Fintype.card V) :
    ∃ f i : ℕ,
      Fintype.card V + f = (letI : DecidableRel L.Adj := dL; L.edgeFinset.card) + 2 ∧
      2 * (letI : DecidableRel L.Adj := dL; L.edgeFinset.card) ≤ i ∧
      3 * f ≤ i := 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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