Euler edge bound for connected planar graphs

Proved

WilliamsonShmoys.connected_planar_edge_bound__p2m_1450b57d_ff3f_4f0e_9779_6941f1e67e4b_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 the specified crossing-free drawing has at most three times its number of vertices minus six edges. The standard proof uses Euler’s formula and the fact that every face boundary has length at least three.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.connected_planar_edge_bound__p2m_1450b57d_ff3f_4f0e_9779_6941f1e67e4b_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hconnected : K.Connected)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn K) :
    letI : ∀ G : SimpleGraph V, DecidableRel G.Adj := fun G => Classical.decRel G.Adj
    K.edgeFinset.card ≤ 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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