Edge-maximal planar graphs satisfy the Euler edge lower bound
ProvedWilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_73c11356_224a_46ad_adf9_93b5e9cc00b6_a1approximation-algorithmsplanar-graphstreewidth
An edge-maximal simple planar graph on a finite linearly ordered vertex type with at least three vertices has at least 3n−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_73c11356_224a_46ad_adf9_93b5e9cc00b6_a1
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V)
(hM : WilliamsonShmoys.HasPlanarDrawingOn M)
(hmax : ∀ N : SimpleGraph V, M ≤ N → WilliamsonShmoys.HasPlanarDrawingOn N → N = M)
(hcard : 3 ≤ Fintype.card V) :
3 * Fintype.card V - 6 ≤ M.edgeSet.ncard := 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).