Edge bound from Euler and face incidence counts

Proved

WilliamsonShmoys.planar_face_count_certificate_edge_bound__p2m_8c314bc4_0fdc_4634_a839_40069b48478f_a2

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

approximation-algorithmsplanar-graphstreewidth

The Euler and triangular incidence equations imply the requested edge lower bound for at least three vertices.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_face_count_certificate_edge_bound__p2m_8c314bc4_0fdc_4634_a839_40069b48478f_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V)
    (F : ℕ) :
    letI : DecidableRel P.Adj := Classical.decRel P.Adj
    (Fintype.card V + F = P.edgeFinset.card + 2) ->
    (3 * F = 2 * P.edgeFinset.card) ->
    (3 <= Fintype.card V) ->
    3 * Fintype.card V - 6 <= P.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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