Euler edge bound for a connected planar simple graph
ProvedWilliamsonShmoys.connected_planar_edge_bound__p2m_8eefeb5d_6bb8_407d_b469_57becc19650a_a1approximation-algorithmsplanar-graphstreewidth
A connected finite simple graph with at least three vertices and a drawing satisfying the target’s crossing-free arc conditions has at most 3|V|−6 edges. This is the source-faithful connected case of the Euler bound and supplies the main combinatorial-topological result. A proof route establishes Euler’s formula for the embedding and counts at least three edge-sides per face, with bridges counted twice; it then translates the face-degree inequality into the edge bound. The condition |V|≥3 is essential to the simple planar bound, and connectedness is used for the Euler formula in this form.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.connected_planar_edge_bound__p2m_8eefeb5d_6bb8_407d_b469_57becc19650a_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hConnected : G.Connected)
(hG : HasPlanarDrawingOn G)
(hcard : 3 <= Fintype.card V) :
(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).