Euler upper bound for drawable simple graphs
ProvedWilliamsonShmoys.planar_edge_count_upper_bound__p2m_4f438a2b_7c24_414a_8cb6_3aa4d799f4d5_a1approximation-algorithmsplanar-graphstreewidth
A drawable simple graph on at least three vertices has at most three times its number of vertices minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_edge_count_upper_bound__p2m_4f438a2b_7c24_414a_8cb6_3aa4d799f4d5_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hG : HasPlanarDrawingOn G)
(hV : 3 ≤ Fintype.card V) :
(letI : DecidableRel G.Adj := Classical.decRel G.Adj
letI : Fintype ↑G.edgeSet := Fintype.ofFinite _
G.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).