Euler edge bound for connected graphs with the given drawing

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WilliamsonShmoys_connected_planar_edge_bound__p2m_e9a12d61_586c_475e_b65b_75c2a4e2b868_a1

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A connected finite simple graph with at least three vertices and a drawing satisfying the exact custom arc conditions has at most 3n−6 edges. This isolates the Euler-formula and face-incidence argument, including the treatment of bridges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys_connected_planar_edge_bound__p2m_e9a12d61_586c_475e_b65b_75c2a4e2b868_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hconn : L.Connected)
    (hL : WilliamsonShmoys.HasPlanarDrawingOn L) :
    L.edgeSet.ncard ≤ 3 * Fintype.card V - 6 := 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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