Euler face count from a crossing-free drawing

Proved

WilliamsonShmoys.planar_drawing_euler_face_count__p2m_ce2523b8_2817_4847_92d2_52162bbbf3f4_a1

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For a finite connected graph with the supplied crossing-free drawing, the number of vertices, edges, and complementary faces satisfies Euler’s formula V − E + F = 2, expressed without subtraction as V + F = E + 2.

Preamble

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

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_euler_face_count__p2m_ce2523b8_2817_4847_92d2_52162bbbf3f4_a1 {V : Type} [Fintype V] [LinearOrder V] (M : SimpleGraph V) [DecidableRel M.Adj] (hM : WilliamsonShmoys.HasPlanarDrawingOn M) (hconn : M.Connected) : ∃ 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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