Triangular face incidence count

Proved

WilliamsonShmoys.maximal_planar_face_incidence__p2m_e77a557c_0981_440f_93d6_d7c4c9434599_a1

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

approximation-algorithmsplanar-graphstreewidth

For any face count satisfying Euler’s equation for this maximal drawing, triangular faces and two sides per edge give three face sides per face and two per edge.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_face_incidence__p2m_e77a557c_0981_440f_93d6_d7c4c9434599_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hK : HasPlanarDrawingOn K)
    (hmax : ∀ L : SimpleGraph V, K ≤ L → HasPlanarDrawingOn L → L ≤ K)
    (f : ℕ) (heuler : Fintype.card V + f = K.edgeSet.ncard + 2) :
    3 * f = 2 * K.edgeSet.ncard := 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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