Euler formula for connected drawable graphs

Proved

WilliamsonShmoys.planar_euler_formula__p2m_ab3c4834_8b15_4a63_8fca_25572aaa5eac_a2

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

approximation-algorithmsplanar-graphstreewidth

For a connected finite graph with a crossing-free drawing, the number of complementary faces satisfies Euler's equation. This child removes the Euler-characteristic calculation from the parent. A proof route is to derive a finite plane embedding from the drawing, define faces as the cycles of its dart permutation, and prove the Euler characteristic formula for a connected rotation system.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_euler_formula__p2m_ab3c4834_8b15_4a63_8fca_25572aaa5eac_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hM : HasPlanarDrawingOn M)
    (hconn : M.Connected) :
    letI : ∀ K : SimpleGraph V, DecidableRel K.Adj :=
      fun K => Classical.decRel K.Adj
    ∃ f : ℕ, Fintype.card V + f = M.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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