GraphConjecture32.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 1 contain sorry, 0 declared as axioms

Written on the Wall II - Conjecture 32

Reference: E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc

Imports (1)

FormalConjecturesUtil

theorem WrittenOnTheWallII.GraphConjecture32.conjecture32 WOWII Conjecture 32 For a simple connected graph sorry line 53 source
Source documentation
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."

L4DEL explanation
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."
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, False, Fintype, DecidableEq, Nontrivial, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.Connected, Finset, Finset.filter, Eq, Nat, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet, SimpleGraph.minDegree, Finset.univ, SimpleGraph.maxDegree and 16 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
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)
…
```
example WrittenOnTheWallII.GraphConjecture32.(example at line 67) line 67 source
Source documentation
The path G invariant cast to ℝ is nonneg.
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ (path G : ℝ) := Nat.cast_nonneg _
…
```
example WrittenOnTheWallII.GraphConjecture32.(example at line 71) line 71 source
Source documentation
In K₃, the max degree is 2.
```
example : (⊤ : SimpleGraph (Fin 3)).maxDegree = 2 := by decide
…
```