GraphConjecture141.lean`

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3 declarations, 0 contain sorry, 0 declared as axioms

Written on the Wall II - Conjecture 141

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

Imports (1)

FormalConjecturesUtil

theorem WrittenOnTheWallII.GraphConjecture141.conjecture141 WOWII Conjecture 141 For a simple connected graph line 45 source
Source documentation
WOWII Conjecture 141
For a simple connected graph $G$, $\mathrm{tree}(G) \ge (1/2) \cdot \mathrm{girth}(G) - 1 + \max_v l(v)$ where $\mathrm{tree}(G)$ is the number of vertices of a largest induced tree subgraph, $\mathrm{girth}(G)$ is the length of the shortest cycle ($0$ if acyclic), and $l(v)$ (indepNeighborsCard G v) is the independence number of the neighbourhood of $v$.

L4DEL explanation
WOWII Conjecture 141 For a simple connected graph $G$, $\mathrm{tree}(G) \ge (1/2) \cdot \mathrm{girth}(G) - 1 + \max_v l(v)$ where $\mathrm{tree}(G)$ is the number of vertices of a largest induced tree subgraph, $\mathrm{girth}(G)$ is the length of the shortest cycle ($0$ if acyclic), and $l(v)$ (indepNeighborsCard G v) is the independence number of the neighbourhood of $v$.
Sources and attribution: Pinned Lean source (1646ca16afd6)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Nontrivial, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.Connected, LE.le, Real, HAdd.hAdd, HSub.hSub, HDiv.hDiv, SimpleGraph.girth, OfNat.ofNat, Nat, Finset.sup, Finset.univ, SimpleGraph.indepNeighborsCard, SimpleGraph.largestInducedTreeSize
Proof uses: And, Membership.mem, Finset, Eq, Exists, Finset.exists_mem_eq_sup, Finset.univ_nonempty, Eq.mpr, id, congrArg, SimpleGraph.indepNeighborsCard_add_one_le_largestInducedTreeSize, HMul.hMul, dite, SimpleGraph.IsAcyclic, Classical.propDecidable, SimpleGraph.IsAcyclic.girth_eq_zero, Decidable.byContradiction, Not, SimpleGraph.three_le_girth, LT.lt and 91 more
129 instances, coercions and auxiliary constants left out.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture141 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
    (G.girth : ℝ) / 2 - 1 + ((Finset.univ.sup (indepNeighborsCard G) : ℕ) : ℝ) ≤
    (largestInducedTreeSize G : ℝ) := by
…
```
example WrittenOnTheWallII.GraphConjecture141.(example at line 76) line 76 source
Source documentation
The largestInducedTreeSize invariant is a natural number (nonneg).
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ largestInducedTreeSize G := Nat.zero_le _
…
```
example WrittenOnTheWallII.GraphConjecture141.(example at line 80) line 80 source
Source documentation
The path graph P₃ has 3 vertices; n P₃ = 3.
```
example : Fintype.card (Fin 3) = 3 := by decide
…
```