Derive the planar edge bound from Euler and face degrees
ProvedWilliamsonShmoys.planar_edge_bound_from_face_count__p2m_c508f2ea_079b_4849_b4cc_0b3b748b75aa_a1approximation-algorithmsplanar-graphstreewidth
For natural vertex, edge, and face counts, the Euler relation and lower bound on total facial degree imply the planar edge bound.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_edge_bound_from_face_count__p2m_c508f2ea_079b_4849_b4cc_0b3b748b75aa_a1
(n e f : Nat)
(hn : 3 ≤ n)
(heuler : e + 2 = n + f)
(hdegree : 3 * f ≤ 2 * e) :
e ≤ 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).