Arithmetic edge bound from Euler and face incidences
ProvedWilliamsonShmoys.nat_euler_face_incidence_edge_bound__p2m_41a70d26_a66a_4916_a2c6_308943ffd553_a1approximation-algorithmsplanar-graphstreewidth
For natural numbers n, m, and f, if n is at least three, n + f = m + 2, and 3f ≤ 2m, then m ≤ 3n − 6. This removes the arithmetic from the parent: combine the equation and incidence inequality to obtain m + 6 ≤ 3n, then convert to the truncated natural subtraction bound. It is independently provable by natural-number arithmetic.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.nat_euler_face_incidence_edge_bound__p2m_41a70d26_a66a_4916_a2c6_308943ffd553_a1
(n m f : Nat) (hn : 3 ≤ n)
(heuler : n + f = m + 2)
(hinc : 3 * f ≤ 2 * m) :
m ≤ 3 * n - 6 := 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).