Face-count identities for maximal planar drawings

Proved

WilliamsonShmoys.maximal_planar_face_count_identities__p2m_24cd5df2_f06e_484a_a9e0_671b5163ec2a_a1

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

approximation-algorithmsplanar-graphstreewidth

For a maximal planar graph on at least three vertices, there is a face count f such that each face contributes three edge incidences and Euler’s formula holds: twice the edge count is three times f, and vertices plus faces equal edges plus two.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_face_count_identities__p2m_24cd5df2_f06e_484a_a9e0_671b5163ec2a_a1 {V : Type} [Fintype V] [LinearOrder V] (M : SimpleGraph V) (hplanar : WilliamsonShmoys.HasPlanarDrawingOn M) (hcard : 3 <= Fintype.card V) (hmax : forall M' : SimpleGraph V, M <= M' -> WilliamsonShmoys.HasPlanarDrawingOn M' -> M' = M) : ∃ f : ℕ, 2 * M.edgeSet.ncard = 3 * f ∧ Fintype.card V + f = M.edgeSet.ncard + 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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