/-
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 32
namespace WrittenOnTheWallII.GraphConjecture32
open Classical SimpleGraph
variable {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α]
WOWII Conjecture 32
For a simple connected graph $G$, $\operatorname{path}(G) \ge \operatorname{dist}_{\operatorname{avg}}(A) + 0.5 \cdot \operatorname{ecc}_{\operatorname{avg}}(M)$, where $\operatorname{path}(G)$ is the floor of the average distance of $G$, $A$ is the set of minimum-degree vertices, $M$ is the set of maximum-degree vertices, $\operatorname{dist}_{\operatorname{avg}}(A)$ is the average distance from all vertices to $A$, and $\operatorname{ecc}_{\operatorname{avg}}(M)$ is the average eccentricity of the vertices in $M$.
The conjecture is false, the authors present a counterexample: "The path on 5 vertices is a counterexample, path = 5, distavg(A) = 4 and the average of eccentricity of maximum degree vertices is 8/3."
@[category research solved, AMS 5]
theorem conjecture32 : answer(False) ↔
∀ (α : Type) [Fintype α] [DecidableEq α] [Nontrivial α]
(G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected),
let A : Finset α := Finset.univ.filter (fun v => G.degree v = G.minDegree)
let M : Finset α := Finset.univ.filter (fun v => G.degree v = G.maxDegree)
let eccavg (S : Finset α) : ℝ := (∑ v ∈ S, (G.eccent v).toNat) / (S.card : ℝ)
distavg G A + (1 / 2 : ℝ) * eccavg M ≤ (path G : ℝ) := ⊢ False ↔
∀ (α : Type) [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α)
[inst_3 : DecidableRel G.Adj],
G.Connected →
let A := {v | G.degree v = G.minDegree};
let M := {v | G.degree v = G.maxDegree};
let eccavg := fun S => ↑(∑ v ∈ S, (G.eccent v).toNat) / ↑S.card;
G.distavg ↑A + 1 / 2 * eccavg M ≤ ↑G.path
All goals completed! 🐙
-- Sanity checks
/-- The `path G` invariant cast to ℝ is nonneg. -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ (path G : ℝ) := Nat.cast_nonneg _
/-- In `K₃`, the max degree is 2. -/
@[category test, AMS 5]
example : (⊤ : SimpleGraph (Fin 3)).maxDegree = 2 := α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ ⊤.maxDegree = 2 All goals completed! 🐙
end WrittenOnTheWallII.GraphConjecture32