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module
public import FormalConjecturesUtilWritten on the Wall II - Conjecture 32
Reference: E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc
@[expose] public sectionnamespace WrittenOnTheWallII.GraphConjecture32open SimpleGraphvariable {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 number of vertices of a largest induced path 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 of all nonzero distances $\operatorname{dist}_G(u, v)$ with $u, v \in A$ (i.e. the average over ordered pairs of distinct vertices of $A$; it is taken to be $0$ when $A$ has fewer than two vertices), 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 distavgA : ℝ := (∑ p ∈ A.offDiag, (G.dist p.1 p.2 : ℝ)) / (A.offDiag.card : ℝ)
let eccavg (S : Finset α) : ℝ := (∑ v ∈ S, (G.eccent v).toNat) / (S.card : ℝ)
distavgA + (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 distavgA := (∑ p ∈ A.offDiag, ↑(G.dist p.1 p.2)) / ↑A.offDiag.card;
let eccavg := fun S ↦ ↑(∑ v ∈ S, (G.eccent v).toNat) / ↑S.card;
distavgA + 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