Derive the planar edge bound from facial counts
ProvedWilliamsonShmoys.planar_facial_count_arithmetic__p2m_6cba1714_c930_4f88_92aa_317652cac9b2_a1approximation-algorithmsplanar-graphstreewidth
If natural numbers for vertices, edges, and faces satisfy Euler's equation and the triangular face-edge incidence equation, then the edge count is at least three times the vertex count minus six.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_facial_count_arithmetic__p2m_6cba1714_c930_4f88_92aa_317652cac9b2_a1
(n m f : ℕ)
(heuler : n + f = m + 2)
(htri : 3 * f = 2 * m) :
3 * n ≤ m + 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).