Planar simple graph edge bound
ProvedWilliamsonShmoys.maximal_planar_graph_edge_bound__p2m_b3b14217_54f7_4da9_9b6d_f0fddbb566d2_a1approximation-algorithmsplanar-graphstreewidth
Every drawable simple graph on at least three vertices has at most three times its number of vertices minus six edges. The Lean form uses the equivalent natural-number inequality edgeSet.ncard + 6 ≤ 3 * card V. Prove this using Euler’s formula and the lower bound of three on face-boundary lengths, with the small-vertex case handled under the stated hypothesis.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_graph_edge_bound__p2m_b3b14217_54f7_4da9_9b6d_f0fddbb566d2_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
(hN : WilliamsonShmoys.HasPlanarDrawingOn N) :
N.edgeSet.ncard + 6 ≤ 3 * Fintype.card V := 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).