Euler edge bound for a connected drawn graph

Proved

WilliamsonShmoys.planar_connected_edge_bound__p2m_4414481d_333a_4f96_8a2d_2d6fda1fe3c0_a1

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A connected finite simple graph with a crossing-free drawing and at least three vertices has at most 3n − 6 edges. Prove this using Euler’s relation and the lower bound of three on each face boundary length.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_connected_edge_bound__p2m_4414481d_333a_4f96_8a2d_2d6fda1fe3c0_a1 {V : Type} [Fintype V] [LinearOrder V] (L : SimpleGraph V) (hDraw : WilliamsonShmoys.HasPlanarDrawingOn L) (hConn : L.Connected) (hV : 3 ≤ Fintype.card V) : L.edgeSet.ncard + 6 ≤ 3 * Fintype.card V := by sorry

Source

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).

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