Edge bound for a connected simple planar graph

Proved

WilliamsonShmoys.planar_edge_bound__p2m_ce11034e_9d40_4f95_bd08_a8f0a5d53ed0_a1

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A connected simple graph on at least three vertices with the supplied crossing-free drawing has at most 3|V|−6 edges. A proof route is to obtain face boundary walks from the embedding, count each edge side twice, use that every such boundary has length at least three under these hypotheses, and apply Euler’s formula. The vertex lower bound is essential to rule out the exceptional one-edge boundary case.

Preamble

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

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_edge_bound__p2m_ce11034e_9d40_4f95_bd08_a8f0a5d53ed0_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 + 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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