Euler face count from a planar drawing

Proved

WilliamsonShmoys.planarDrawingOn_eulerFormula__p2m_da53f01a_4759_4677_9c66_1d37b711de18_a1

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

approximation-algorithmsplanar-graphstreewidth

A crossing-free drawing of a finite connected graph gives a face count satisfying Euler’s formula.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planarDrawingOn_eulerFormula__p2m_da53f01a_4759_4677_9c66_1d37b711de18_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (dL : DecidableRel L.Adj)
    (hL : HasPlanarDrawingOn L)
    (hconn : L.Connected) :
    ∃ f : ℕ,
      Fintype.card V + f = (letI : DecidableRel L.Adj := dL; L.edgeFinset.card) + 2 := 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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