/-
Copyright 2026 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
import FormalConjecturesUtilWritten on the Wall II - Conjecture 36
namespace WrittenOnTheWallII.GraphConjecture36
open Classical SimpleGraph
dp G is the number of diametrical pairs of G: the number of unordered
pairs {u, v} of vertices at distance diam(G).
noncomputable def dp {α : Type*} [Fintype α] (G : SimpleGraph α) : ℕ :=
(Finset.univ.filter
(fun p : Sym2 α => p.lift ⟨fun u v => G.dist u v = G.diam ∧ u ≠ v,
fun u v => α:Type u_1inst✝:Fintype αG:SimpleGraph αp:Sym2 αu:αv:α⊢ (fun u v => G.dist u v = G.diam ∧ u ≠ v) u v = (fun u v => G.dist u v = G.diam ∧ u ≠ v) v u All goals completed! 🐙⟩)).card
WOWII Conjecture 36:
For every finite simple connected graph $G$,
$\operatorname{path}(G) \ge 2 \cdot \operatorname{rad}(G) / \operatorname{dp}(G)$,
where $\operatorname{path}(G)$ is the floor of the average distance of $G$,
$\operatorname{rad}(G)$ is the radius of $G$, and $\operatorname{dp}(G)$ is the number
of
Disproved by Waller in Oct 2003 (counterexample: path number 5, radius 3, dp 1).
@[category research solved, AMS 5]
theorem conjecture36 : answer(False) ↔
∀ {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α],
∀ (G : SimpleGraph α) [DecidableRel G.Adj] (_ : G.Connected) (_ : 0 < dp G),
(2 * G.radius.toNat : ℝ) / (dp G : ℝ) ≤ (path G : ℝ) := ⊢ False ↔
∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj],
G.Connected → 0 < dp G → 2 * ↑G.radius.toNat / ↑(dp G) ≤ ↑G.path
All goals completed! 🐙
-- Sanity checks
/-- The `path G` invariant is nonneg. -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ (path G : ℝ) := Nat.cast_nonneg _
end WrittenOnTheWallII.GraphConjecture36