Facial bound from Euler’s identity and the planar edge bound
ProvedWilliamsonShmoys.planar_face_bound_from_euler_and_edge_bound__p2m_7b82f13f_4972_4c18_a653_c5d0801ea028_a2approximation-algorithmsplanar-graphstreewidth
For natural numbers E, V, and f, if E + 2 = V + f and E + 6 ≤ 3V, then 3f ≤ 2E.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_face_bound_from_euler_and_edge_bound__p2m_7b82f13f_4972_4c18_a653_c5d0801ea028_a2
(E V f : Nat)
(heuler : E + 2 = V + f)
(hbound : E + 6 <= 3 * V) :
3 * f <= 2 * E := 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).