Maximum edge count in a nonempty family of simple graphs

Proved

WilliamsonShmoys.exists_max_card_simpleGraph__p2m_15de7f60_3f94_44ed_9413_540d275804dc_a3

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

approximation-algorithmsplanar-graphstreewidth

Every nonempty family of simple graphs on a finite vertex type with decidable equality contains a graph whose edge count is at least that of every other member.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_max_card_simpleGraph__p2m_15de7f60_3f94_44ed_9413_540d275804dc_a3
    {V : Type} [Fintype V] [DecidableEq V]
    (P : SimpleGraph V → Prop) (G₀ : SimpleGraph V) (h₀ : P G₀) :
    ∃ G : SimpleGraph V, P G ∧
      (letI : DecidableRel G.Adj := Classical.decRel G.Adj
       ∀ H : SimpleGraph V, P H →
         letI : DecidableRel H.Adj := Classical.decRel H.Adj
         H.edgeFinset.card ≤ G.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).

View graph