Derive finite face data from the maximal planar edge count

Open

WilliamsonShmoys.planarMaximalEdgeCount_to_PlanarMapData__p2m_b3eca2bb_f78b_4222_b8fc_fe99bf34c163_a1

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For a finite simple graph with at least three vertices and 3|V|−6 edges, there is finite face data satisfying the specified Euler equation and face-degree equations.

Preamble

import Definitions.Def_PlanarMapData__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a3
import Mathlib

Formal statement

theorem WilliamsonShmoys.planarMaximalEdgeCount_to_PlanarMapData__p2m_b3eca2bb_f78b_4222_b8fc_fe99bf34c163_a1
    {V : Type} [Fintype V]
    (J : SimpleGraph V) [DecidableRel J.Adj]
    (hcard : 3 <= Fintype.card V)
    (hedge : J.edgeFinset.card = 3 * Fintype.card V - 6) :
    Nonempty (@PlanarMapData__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a3 V (inferInstance) J inferInstance) := 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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