Edge bound from Euler and face data
ProvedWilliamsonShmoys.euler_data_edge_bound__p2m_b0556792_8fbb_42b1_83f9_e49450aba15a_a2approximation-algorithmsplanar-graphstreewidth
For natural-number vertex, edge, face, and total-boundary counts, Euler's equation, two sides per edge, and the total face-length lower bound imply the planar edge bound. This child removes the arithmetic from the parent and has an independent proof by substituting the equalities and using natural-number linear arithmetic. The hypothesis n ≥ 3 is retained from the drawing argument, although the displayed arithmetic implication itself follows without it.
Preamble
import Mathlib
Formal statement
theorem WilliamsonShmoys.euler_data_edge_bound__p2m_b0556792_8fbb_42b1_83f9_e49450aba15a_a2
(n e f l : ℕ)
(hn : 3 ≤ n)
(hEuler : n + f = e + 2)
(hSides : l = 2 * e)
(hFaces : 3 * f ≤ l) :
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).