Euler upper bound for drawable simple graphs
ProvedWilliamsonShmoys.planar_edge_count_upper_bound__p2m_9518e4d8_9181_45e5_8d65_e8b1bfbc442a_a1approximation-algorithmsplanar-graphstreewidth
Every finite drawable simple graph with at least three vertices has at most 3|V|−6 edges. This is the Euler edge bound; the lower-size hypothesis is retained for its boundary cases.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_edge_count_upper_bound__p2m_9518e4d8_9181_45e5_8d65_e8b1bfbc442a_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hcard : 3 <= Fintype.card V)
(hG : WilliamsonShmoys.HasPlanarDrawingOn G) :
G.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).