Euler edge bound for a connected planar graph

Proved

WilliamsonShmoys.connected_planar_euler_bound__p2m_fa5d736f_5e40_4868_bcf6_007f98212d2a_a1

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

approximation-algorithmsplanar-graphstreewidth

For a connected graph with the target’s crossing-free drawing and at least three vertices, prove the Euler edge bound. This isolates face counting from the parent’s disconnected case. A proof route is to form the plane embedding’s faces, use Euler’s relation, count each edge twice in face boundaries, and show each face boundary has length at least three using simplicity and the vertex-count hypothesis. The remaining core obligation is formalizing those face and boundary counts from the given arc-based drawing.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.connected_planar_euler_bound__p2m_fa5d736f_5e40_4868_bcf6_007f98212d2a_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hconn : H.Connected)
    (hcard : 3 ≤ Fintype.card V) :
    letI : DecidableRel H.Adj := Classical.decRel H.Adj
    H.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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