Euler edge bound for connected planar graphs
OpenWilliamsonShmoys.connected_planar_drawing_edge_bound__p2m_b05b2ac9_cf86_42cd_81eb_d15df4254060_a1approximation-algorithmsplanar-graphstreewidth
A connected finite simple graph with a crossing-free drawing and at least three vertices has at most 3n−6 edges. A proof route is Euler’s formula together with a treatment of bridges or an edge-maximal planar extension.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.connected_planar_drawing_edge_bound__p2m_b05b2ac9_cf86_42cd_81eb_d15df4254060_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
(hConn : H.Connected) (hH : HasPlanarDrawingOn H) :
(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).