Euler upper bound for planar graph edges
ProvedWilliamsonShmoys.edge_maximal_planar_graph_edge_count_upper__p2m_401bb865_eb7f_4687_8488_019108bab1f9_a1approximation-algorithmsplanar-graphstreewidth
Every finite simple graph with a crossing-free drawing and at least three vertices has at most 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.edge_maximal_planar_graph_edge_count_upper__p2m_401bb865_eb7f_4687_8488_019108bab1f9_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V)
(hN : WilliamsonShmoys.HasPlanarDrawingOn N)
(hcard : 3 <= Fintype.card V) :
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).