Edge bound for a connected simple planar graph

Proved

WilliamsonShmoys.planar_simple_graph_edge_bound__p2m_a18184f7_c783_4d00_9241_7e8764df5618_a1

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

approximation-algorithmsplanar-graphstreewidth

A connected simple graph with the supplied planar drawing and at least three vertices has at most 3n − 6 edges. A proof route is to use the face-boundary count, where every face boundary has length at least three and every edge contributes two sides, then combine that count with Euler’s formula. Bridges contribute both sides to their boundary walks.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_simple_graph_edge_bound__p2m_a18184f7_c783_4d00_9241_7e8764df5618_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) [DecidableRel M.Adj]
    (hM : WilliamsonShmoys.HasPlanarDrawingOn M)
    (hconn : M.Connected)
    (hcard : 3 ≤ Fintype.card V) :
    (M.edgeFinset.card : ℤ) ≤ 3 * (Fintype.card V : ℤ) - 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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