Euler and triangular-face equations for a maximal planar drawing

Proved

WilliamsonShmoys.maximal_planar_face_equations__p2m_a641d14b_2083_49f9_8715_0f20d0b3e1ec_a1

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

approximation-algorithmsplanar-graphstreewidth

For a maximal drawable simple graph on at least three vertices, there is a face count satisfying Euler’s equation and the triangular face-edge incidence equation. The first equation counts vertices, faces, and edges; the second expresses that every face has three boundary edges and every edge is incident to two faces.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

Formal statement

theorem WilliamsonShmoys.maximal_planar_face_equations__p2m_a641d14b_2083_49f9_8715_0f20d0b3e1ec_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hmax : ∀ L : SimpleGraph V,
      WilliamsonShmoys.HasPlanarDrawingOn L → H ≤ L → L = H) :
    (letI : DecidableRel H.Adj := Classical.decRel H.Adj
     ∃ f : ℕ,
      Fintype.card V + f = H.edgeFinset.card + 2 ∧
        3 * f = 2 * H.edgeFinset.card) := 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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