The planar target count fits within the complete graph

Proved

WilliamsonShmoys.planar_edge_count_capacity__p2m_f3a63840_9387_418b_904c_fbe2e8408361_a1

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

approximation-algorithmsplanar-graphstreewidth

For n≥3, the target edge count 3n−6 is at most the number n(n−1)/2 of unordered pairs of distinct vertices.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_edge_count_capacity__p2m_f3a63840_9387_418b_904c_fbe2e8408361_a1
    (n : Nat) (hn : 3 ≤ n) :
    3 * n - 6 ≤ n * (n - 1) / 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).

View graph