Planar graphs have at most three n minus six edges
ProvedWilliamsonShmoys.planar_edge_count_upper_bound__p2m_2e383da1_b0fd_42ac_bf20_409641577930_a2approximation-algorithmsplanar-graphstreewidth
Every finite simple graph on an ordered vertex type with at least three vertices and a drawing satisfying HasPlanarDrawingOn has at most 3n−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_edge_count_upper_bound__p2m_2e383da1_b0fd_42ac_bf20_409641577930_a2
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hdraw : WilliamsonShmoys.HasPlanarDrawingOn K) :
letI : ∀ G : SimpleGraph V, DecidableRel G.Adj := fun G => Classical.decRel G.Adj
K.edgeFinset.card ≤ 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).