Euler face-count inequalities imply the planar edge bound
ProvedWilliamsonShmoys.planar_euler_edge_arithmetic__p2m_00ee0ef8_811a_4a39_b8c0_1ed175b9b4ef_a1approximation-algorithmsplanar-graphstreewidth
If the Euler equation holds for vertex, edge, and face counts and the total face-boundary length is at most twice the edge count, then the edge count is at most 3n − 6. The hypothesis n ≥ 3 is preserved; this child is a direct natural-number arithmetic step.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_euler_edge_arithmetic__p2m_00ee0ef8_811a_4a39_b8c0_1ed175b9b4ef_a1
(n m f : ℕ)
(hn : 3 ≤ n)
(heuler : n + f = m + 2)
(hboundary : 3 * f ≤ 2 * m) :
m + 6 ≤ 3 * n := by sorrySource
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).