Edge bound from Euler and triangular-face counts
ProvedWilliamsonShmoys.euler_triangulation_edge_arithmetic__p2m_306301a2_72f6_4d8f_a2c9_6046e69fec66_a1approximation-algorithmsplanar-graphstreewidth
For natural numbers n, e, and f, Euler’s equation and the triangular-face incidence equation imply the planar edge bound. The proof is integer algebra; the lower bound on n records the target’s boundary hypothesis.
Preamble
import Mathlib
Formal statement
theorem WilliamsonShmoys.euler_triangulation_edge_arithmetic__p2m_306301a2_72f6_4d8f_a2c9_6046e69fec66_a1
(n e f : ℕ)
(hn : 3 <= n)
(hEuler : (n : ℤ) - (e : ℤ) + (f : ℤ) = 2)
(htri : 3 * f = 2 * e) :
e + 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).