GraphConjecture17.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 17

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

Imports (1)

FormalConjecturesUtil

theorem WrittenOnTheWallII.GraphConjecture17.conjecture17 Documented declaration: conjecture17 sorry line 44 source
Source documentation
WOWII Conjecture 17
For a simple connected graph G, the size b(G) of a largest induced bipartite subgraph satisfies b(G) ≥ α(G) + ⌈diam(G) / 3⌉, where α(G) is the independence number of G and diam(G) is the diameter of G.

L4DEL explanation
WOWII Conjecture 17 For a simple connected graph G, the size b(G) of a largest induced bipartite subgraph satisfies b(G) ≥ α(G) + ⌈diam(G) / 3⌉, where α(G) is the independence number of G and diam(G) is the diameter of G.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Fintype, DecidableEq, Nontrivial, SimpleGraph, SimpleGraph.Connected, LE.le, Real, HAdd.hAdd, SimpleGraph.indepNum, Int.ceil, HDiv.hDiv, SimpleGraph.diam, OfNat.ofNat, Nat, SimpleGraph.b
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture17 (G : SimpleGraph α) (h : G.Connected) :
    (G.indepNum : ℝ) + ⌈(G.diam : ℝ) / 3⌉ ≤ b G := by
…
```
example WrittenOnTheWallII.GraphConjecture17.(example at line 52) line 52 source
Source documentation
The invariant b G is nonneg (it's the cast of a natural number).
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ b G := Nat.cast_nonneg _
…
```
example WrittenOnTheWallII.GraphConjecture17.(example at line 56) line 56 source
Source documentation
The independence number α(K₂) equals 1 (each independent set contains at most one vertex).
```
example : (⊤ : SimpleGraph (Fin 2)).edgeFinset.card = 1 := by decide
…
```