GraphConjecture13.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 13

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

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

theorem WrittenOnTheWallII.GraphConjecture13.conjecture13 Documented declaration: conjecture13 sorry line 45 source
Source documentation
WOWII Conjecture 13
For a simple connected graph G, the size b(G) of a largest induced bipartite subgraph satisfies b(G) ≥ diam(G) + max_{v ∈ V} l(v) - 1, where diam(G) is the diameter of G and l(v) = indepNeighborsCard G v is the independence number of the neighbourhood of v.

L4DEL explanation
WOWII Conjecture 13 For a simple connected graph G, the size b(G) of a largest induced bipartite subgraph satisfies b(G) ≥ diam(G) + max_{v ∈ V} l(v) - 1, where diam(G) is the diameter of G and l(v) = indepNeighborsCard G v is the independence number of the neighbourhood of v.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Fintype, DecidableEq, Nontrivial, SimpleGraph, SimpleGraph.Connected, LE.le, Real, HSub.hSub, HAdd.hAdd, SimpleGraph.diam, Finset.max', Nat, Finset.image, SimpleGraph.indepNeighborsCard, Finset.univ, OfNat.ofNat, SimpleGraph.b
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
25 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture13 (G : SimpleGraph α) (h : G.Connected) :
    letI maxL := (Finset.univ.image (fun v => indepNeighborsCard G v)).max' (by simp)
…
```
example WrittenOnTheWallII.GraphConjecture13.(example at line 54) line 54 source
Source documentation
The largest induced bipartite subgraph invariant b G is nonneg for any graph.
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ b G := Nat.cast_nonneg _
…
```
example WrittenOnTheWallII.GraphConjecture13.(example at line 59) line 59 source
Source documentation
In K₂, all vertices have degree 1, so indepNeighborsCard at vertex 0 is 0 (the neighbourhood of 0 consists of only vertex 1, which has no independent edges).
```
example : (⊤ : SimpleGraph (Fin 2)).maxDegree = 1 := by decide
…
```