Euler certificate implies the planar edge bound

Proved

WilliamsonShmoys.euler_certificate_edge_bound__p2m_10555cd3_1e32_4780_a0c9_e859a324fd87_a1

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

approximation-algorithmsplanar-graphstreewidth

For natural numbers n, m, m', and f, the stated Euler equation and face-degree inequality imply m' ≤ 3 * n - 6; the bound m ≤ m' then gives the desired edge bound. The hypothesis 3 ≤ n ensures the natural-number subtraction has the intended value. This is a linear arithmetic step.

Preamble

import Mathlib

Formal statement

theorem WilliamsonShmoys.euler_certificate_edge_bound__p2m_10555cd3_1e32_4780_a0c9_e859a324fd87_a1
    (n m m' f : Nat)
    (hn : 3 ≤ n)
    (hcert : m ≤ m' ∧ n + f = m' + 2 ∧ 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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