Planar graphs satisfy the Euler edge bound
ProvedWilliamsonShmoys.planar_edge_count_upper_bound__p2m_b1eeea0d_0264_4037_a9be_26c96369c701_a1approximation-algorithmsplanar-graphstreewidth
Every drawable simple graph on at least three finite vertices has at most 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_edge_count_upper_bound__p2m_b1eeea0d_0264_4037_a9be_26c96369c701_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
K.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).