GraphConjecture3.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 3

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

Imports (1)

FormalConjecturesUtil

theorem WrittenOnTheWallII.GraphConjecture3.conjecture3 Documented declaration: conjecture3 sorry line 46 source
Source documentation
WOWII Conjecture 3
For a connected simple graph G, the number of leaves in a maximum spanning tree satisfies Ls(G) ≥ G.indepDominationNumber * MaxTemp(G), where G.indepDominationNumber is the independent domination number and MaxTemp(G) is max_v deg(v)/(n(G) - deg(v)).

L4DEL explanation
WOWII Conjecture 3 For a connected simple graph G, the number of leaves in a maximum spanning tree satisfies Ls(G) ≥ G.indepDominationNumber * MaxTemp(G), where G.indepDominationNumber is the independent domination number and MaxTemp(G) is max_v deg(v)/(n(G) - deg(v)).
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nontrivial, SimpleGraph.Connected, LE.le, Real, HMul.hMul, SimpleGraph.indepDominationNumber, SimpleGraph.MaxTemp, SimpleGraph.Ls
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture3 {G : SimpleGraph α} [DecidableRel G.Adj] [Nontrivial α] (h_conn : G.Connected) :
    G.indepDominationNumber * MaxTemp G ≤ Ls G := by
…
```
example WrittenOnTheWallII.GraphConjecture3.(example at line 54) line 54 source
Source documentation
The number of vertices of the two-vertex graph K₂ is 2.
```
example : Fintype.card (Fin 2) = 2 := rfl
…
```
example WrittenOnTheWallII.GraphConjecture3.(example at line 58) line 58 source
Source documentation
In K₂, the temperature of vertex 0 is deg(0) / (n - deg(0)) = 1 / 1 = 1.
```
example : temp_v (⊤ : SimpleGraph (Fin 2)) ⟨0, by omega⟩ = 1 := by
…
```