Maximal planar graph edge lower bound
OpenWilliamsonShmoys.maximal_planar_edgeSet_ncard_ge_three_mul_card_sub_six__p2m_b8afd70d_5e18_4708_a739_37d2bdc4e576_a1approximation-algorithmsplanar-graphstreewidth
An edge-maximal graph under the given finite drawing predicate, with at least three vertices, has at least 3|V|−6 edges. Prove by showing maximality forces a triangulated planar embedding, then apply Euler’s formula and count face-edge incidences.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_edgeSet_ncard_ge_three_mul_card_sub_six__p2m_b8afd70d_5e18_4708_a739_37d2bdc4e576_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hplanar : HasPlanarDrawingOn K)
(hmax : ∀ L : SimpleGraph V,
K.edgeSet ⊆ L.edgeSet →
HasPlanarDrawingOn L →
L.edgeSet = K.edgeSet) :
3 * Fintype.card V - 6 ≤ K.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).