Facial boundary lengths at an Euler face count

Proved

WilliamsonShmoys.planar_drawing_boundary_lengths_at_euler_count__p2m_87b40329_c584_43e3_b2a8_a209b09780ca_a2

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

approximation-algorithmsplanar-graphstreewidth

At any natural face count satisfying Euler’s equation, enumerate facial boundary walks so their lengths sum to twice the edge count and every walk has length at least three.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_boundary_lengths_at_euler_count__p2m_87b40329_c584_43e3_b2a8_a209b09780ca_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) [DecidableRel K.Adj]
    (hK : HasPlanarDrawingOn K)
    (hconn : K.Connected)
    (hcard : 3 ≤ Fintype.card V)
    (faceCount : Nat)
    (hEuler : Fintype.card V + faceCount = K.edgeFinset.card + 2) :
    ∃ boundaryLength : Fin faceCount → Nat,
      (∑ f : Fin faceCount, boundaryLength f) = 2 * K.edgeFinset.card ∧
      (∀ f : Fin faceCount, 3 ≤ boundaryLength f) := 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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