Simple planar graphs have at most 3n−6 edges
ProvedWilliamsonShmoys.planar_simple_graph_edge_upper_bound__p2m_2a34f2f7_4575_4fc5_979a_579be47b545a_a1approximation-algorithmsplanar-graphstreewidth
A drawable simple graph on at least three finite linearly ordered vertices has at most 3n−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_simple_graph_edge_upper_bound__p2m_2a34f2f7_4575_4fc5_979a_579be47b545a_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V)
(hN : WilliamsonShmoys.HasPlanarDrawingOn N)
(hcard : 3 ≤ Fintype.card V) :
N.edgeSet.ncard ≤ 3 * Fintype.card V - 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).