Maximal planar edge count lower bound

Proved

WilliamsonShmoys.maximal_planar_edge_count_lower_bound__p2m_0227f8b9_c5e1_4755_9e67_a93be5e3575c_a1

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

approximation-algorithmsplanar-graphstreewidth

A graph with a crossing-free drawing that is maximal under drawable supergraphs on the same vertex type has at least 3|V|−6 edges when |V|≥3. Use maximality to show the drawing's faces can be triangulated only if they are already triangles, then apply Euler's formula and count edge-sides. Together with the upper bound, this yields the target equality.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count_lower_bound__p2m_0227f8b9_c5e1_4755_9e67_a93be5e3575c_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (Q : SimpleGraph V)
    (hcard : 3 <= Fintype.card V)
    (hQ : WilliamsonShmoys.HasPlanarDrawingOn Q)
    (hmax : forall R : SimpleGraph V,
      (forall {u v : V}, Q.Adj u v -> R.Adj u v) ->
      WilliamsonShmoys.HasPlanarDrawingOn R ->
      R = Q) :
    letI : DecidableRel Q.Adj := Classical.decRel Q.Adj
    3 * Fintype.card V <= Q.edgeFinset.card + 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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