Edge count of a maximal planar graph
ProvedWilliamsonShmoys.maximal_planar_edge_count__p2m_6467164a_4ac7_4202_bbd8_0cd7677d3ce4_a2approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal planar simple graph on at least three vertices has exactly 3|V|−6 edges. This removes the edge-count obligation from the parent. A proof route is to show maximality forces connectedness and triangular faces, count incidences between faces and edges, and combine that count with Euler’s formula. At |V| = 3, the graph is K₃ and the equation holds. The face and incidence arguments for the supplied arc-based drawing predicate remain the core obligation.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_6467164a_4ac7_4202_bbd8_0cd7677d3ce4_a2
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
(hmax : forall K : SimpleGraph V,
H <= K -> WilliamsonShmoys.HasPlanarDrawingOn K -> K = H)
(hcard : 3 <= Fintype.card V) :
letI : DecidableRel H.Adj := Classical.decRel H.Adj
H.edgeFinset.card + 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).