GraphConjecture172.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 172

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

The conjecture is false for the graph D₉, formed from two triangles by joining distinguished vertices with a path of nine edges. It has at most four leaves in a spanning tree. Its peripheral vertices have degree two, while the two maximum-degree vertices of its square are at distance seven in the original graph, so the claimed inequality would give 4 ≥ 8.

Imports (1)

FormalConjecturesUtil

def WrittenOnTheWallII.GraphConjecture172.peripheryMaxDegree The maximum original-graph degree among the peripheral vertices line 41 source
Source documentation
The maximum original-graph degree among the peripheral vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nat
Proof uses: Finset.sup, Finset.filter, Membership.mem, Set, SimpleGraph.maxEccentricityVertices, Classical.propDecidable, Finset.univ, Finset.card, SimpleGraph.neighborFinset, SimpleGraph.neighborSet
6 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
noncomputable def peripheryMaxDegree {V : Type} [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj] : ℕ := by
…
```
def WrittenOnTheWallII.GraphConjecture172.squareMaximumDegreeVertices The maximum-degree vertices of G², presented using finite graph distance line 48 source
Source documentation
The maximum-degree vertices of G², presented using finite graph distance.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Set
Proof uses: Nat, Finset.card, Finset.filter, And, Ne, LE.le, SimpleGraph.computable_dist, OfNat.ofNat, Eq, Finset.univ, Set.ofPred, Finset.sup
8 instances, coercions and auxiliary constants left out.
```
def squareMaximumDegreeVertices {V : Type} [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj] : Set V :=
…
```
theorem WrittenOnTheWallII.GraphConjecture172.conjecture172 sorry line 62 source
depends on sorry what this means
Statement uses: Iff, False, Fintype, DecidableEq, Nontrivial, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.Connected, GE.ge, Real, SimpleGraph.Ls, HAdd.hAdd, Neg.neg, OfNat.ofNat, WrittenOnTheWallII.GraphConjecture172.peripheryMaxDegree, SimpleGraph.distMin, WrittenOnTheWallII.GraphConjecture172.squareMaximumDegreeVertices
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.