Euler face count for a connected planar drawing

Proved

WilliamsonShmoys.connected_planar_drawing_euler_formula__p2m_6e5a9dbe_f5b6_4fde_9426_557ef9f6a091_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite connected graph with at least three vertices and a crossing-free drawing has a natural face count satisfying Euler’s formula |V| + f = |E| + 2. Prove by obtaining the finite plane embedding from the drawing and applying the Euler formula for connected plane graphs; the vertex-count lower bound is retained to match the parent.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.connected_planar_drawing_euler_formula__p2m_6e5a9dbe_f5b6_4fde_9426_557ef9f6a091_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hcard : 3 <= Fintype.card V)
    (hconn : K.Connected)
    (hK : HasPlanarDrawingOn K) :
    letI : (L : SimpleGraph V) -> DecidableRel L.Adj :=
      fun L => Classical.decRel L.Adj
    ∃ f : Nat, Fintype.card V + f = K.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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