Euler equation for a connected planar drawing

Proved

WilliamsonShmoys.planar_drawing_euler_equation_data__p2m_c26dc073_3bb3_419b_aef0_cb3d89286f37_a1

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

approximation-algorithmsplanar-graphstreewidth

A finite connected graph with a crossing-free drawing has a natural-number face count satisfying Euler’s equation. The independent proof route is to form the planar map from the drawing, use connectedness to apply Euler’s formula, and identify the map’s vertices and edges with those of the graph. This isolates the Euler/map construction; it does not establish the planar edge bound.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_euler_equation_data__p2m_c26dc073_3bb3_419b_aef0_cb3d89286f37_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hconn : H.Connected)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H) :
    (letI : DecidableRel H.Adj := Classical.decRel H.Adj
     ∃ f : Nat,
       Fintype.card V + f = H.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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