Euler edge bound for connected planar simple graphs
OpenWilliamsonShmoys.connected_planar_graph_edge_bound__p2m_e0c85ed3_29c9_4753_86a9_a6c3e4be15ff_a2approximation-algorithmsplanar-graphstreewidth
A connected finite simple graph with at least three vertices and a crossing-free drawing has at most three times its vertex count minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.connected_planar_graph_edge_bound__p2m_e0c85ed3_29c9_4753_86a9_a6c3e4be15ff_a2
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V)
(hcard : 3 ≤ Fintype.card V)
(hconn : G.Connected)
(hdraw : WilliamsonShmoys.HasPlanarDrawingOn G) :
(letI : DecidableRel G.Adj := Classical.decRel G.Adj
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).