Lower edge-count bound for maximal planar graphs
ProvedWilliamsonShmoys.maximal_planar_edge_count_lower_bound__p2m_c1b4f689_ce4a_40bb_ab72_071998d2bd64_a1approximation-algorithmsplanar-graphstreewidth
An edge-maximal graph with a crossing-free drawing and at least three vertices has at least 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count_lower_bound__p2m_c1b4f689_ce4a_40bb_ab72_071998d2bd64_a1
{V : Type} [Fintype V] [LinearOrder V]
(P : SimpleGraph V)
(hP : WilliamsonShmoys.HasPlanarDrawingOn P)
(hmax : ∀ Q : SimpleGraph V, P ≤ Q →
WilliamsonShmoys.HasPlanarDrawingOn Q → Q = P)
(hcard : 3 ≤ Fintype.card V)
[DecidableRel P.Adj] :
3 * Fintype.card V ≤ P.edgeFinset.card + 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).