Euler edge count for a maximal drawable graph

Proved

WilliamsonShmoys.maximal_drawable_graph_has_euler_edge_count__p2m_c985394f_ff9a_4827_826c_d7a45e5d518a_a2

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

approximation-algorithmsplanar-graphstreewidth

A drawable graph on at least three vertices that has no strictly larger drawable supergraph on the same vertices has exactly three times the number of vertices minus six edges. The independent proof route is to show maximality forces a triangulation and apply Euler’s formula, preserving the three-vertex lower bound.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_drawable_graph_has_euler_edge_count__p2m_c985394f_ff9a_4827_826c_d7a45e5d518a_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (hcard : 3 <= Fintype.card V)
    (hdraw : HasPlanarDrawingOn N)
    (hmax : forall H : SimpleGraph V,
      (forall u v, N.Adj u v -> H.Adj u v) ->
      HasPlanarDrawingOn H -> H = N) :
    letI : forall H : SimpleGraph V, DecidableRel H.Adj := fun H => Classical.decRel H.Adj
    3 * Fintype.card V - 6 = N.edgeFinset.card := 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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