GraphConjecture23.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 23

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

Imports (1)

FormalConjecturesUtil

theorem WrittenOnTheWallII.GraphConjecture23.conjecture23 Documented declaration: conjecture23 sorry line 47 source
Source documentation
WOWII Conjecture 23
For a simple connected graph G, b(G) ≥ ⌊α(G) + dist_avg(M) / 2⌋, where b(G) is the size of a largest induced bipartite subgraph, α(G) is the independence number, M is the set of maximum-degree vertices, and dist_avg(M) is the average distance between distinct vertices of M.
This conjecture is false; there is a counterexample with b(G) = 19, α(G) = 15, and dist_avg(M) = 10.

L4DEL explanation
WOWII Conjecture 23 For a simple connected graph G, b(G) ≥ ⌊α(G) + dist_avg(M) / 2⌋, where b(G) is the size of a largest induced bipartite subgraph, α(G) is the independence number, M is the set of maximum-degree vertices, and dist_avg(M) is the average distance between distinct vertices of M. This conjecture is false; there is a counterexample with b(G) = 19, α(G) = 15, and dist_avg(M) = 10.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, False, Fintype, DecidableEq, Nontrivial, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.Connected, Set, Set.ofPred, Eq, Nat, SimpleGraph.degree, Membership.mem, SimpleGraph.neighborSet, SimpleGraph.maxDegree, LE.le, Real, Int.floor and 6 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture23 : answer(False) ↔
  ∀ (α : Type) [Fintype α] [DecidableEq α] [Nontrivial α]
    (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected),
    let M : Set α := {v | G.degree v = G.maxDegree}
…
```
example WrittenOnTheWallII.GraphConjecture23.(example at line 58) line 58 source
Source documentation
The invariant b G is nonneg.
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ b G := Nat.cast_nonneg _
…
```
example WrittenOnTheWallII.GraphConjecture23.(example at line 62) line 62 source
Source documentation
In K₃, the max degree is 2.
```
example : (⊤ : SimpleGraph (Fin 3)).maxDegree = 2 := by decide
…
```