Edge bound from Euler and face lengths

Proved

planar_edge_bound_of_face_data__p2m_65b9f3b0_4e49_457f_8ae3_e5dab0f54711_a2

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

approximation-algorithmsplanar-graphstreewidth

Natural numbers satisfying n ≥ 3, n + f = m + 2, and 3f ≤ 2m satisfy m ≤ 3n − 6.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem planar_edge_bound_of_face_data__p2m_65b9f3b0_4e49_457f_8ae3_e5dab0f54711_a2
    (n f m : Nat) (hn : 3 ≤ n)
    (heuler : n + f = m + 2)
    (hfaces : 3 * f ≤ 2 * m) :
    m ≤ 3 * n - 6 := 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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