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import FormalConjecturesUtilWritten on the Wall II - Conjecture 65
namespace WrittenOnTheWallII.GraphConjecture65
open Classical SimpleGraph
variable {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α]
WOWII Conjecture 65:
For a simple connected graph $G$, the size $f(G)$ of a largest induced forest satisfies
$f(G) \ge \operatorname{dist_min}(A) + \lceil \operatorname{dist_min}(M) / 3 \rceil$,
where $A$ is the set of minimum-degree vertices, $M$ is the set of maximum-degree vertices,
and $\operatorname{dist_min}(S) = \min_{v \notin S} \operatorname{dist}(v, S)$ (see distMin).
@[category research open, AMS 5]
theorem conjecture65 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
let A : Set α := {v | G.degree v = G.minDegree}
let M : Set α := {v | G.degree v = G.maxDegree}
(distMin G A : ℝ) + ⌈(distMin G M : ℝ) / 3⌉ ≤ (G.largestInducedForestSize : ℝ) := α:Type u_1inst✝³:Fintype αinst✝²:DecidableEq αinst✝¹:Nontrivial αG:SimpleGraph αinst✝:DecidableRel G.Adjh:G.Connected⊢ let A := {v | G.degree v = G.minDegree};
let M := {v | G.degree v = G.maxDegree};
↑(G.distMin A) + ↑⌈↑(G.distMin M) / 3⌉ ≤ ↑G.largestInducedForestSize
All goals completed! 🐙
-- Sanity checks
/-- The `largestInducedForestSize` is nonneg. -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ G.largestInducedForestSize := Nat.zero_le _
/-- In the complete graph `K₃`, min degree equals max degree (regular graph). -/
@[category test, AMS 5]
example : (⊤ : SimpleGraph (Fin 3)).minDegree = (⊤ : SimpleGraph (Fin 3)).maxDegree := α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ ⊤.minDegree = ⊤.maxDegree
All goals completed! 🐙
/-- `distMin G S` is always nonneg. -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) (S : Set (Fin 3)) : 0 ≤ distMin G S := Nat.zero_le _
/-- `distToSet G v S` is always nonneg. -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) (v : Fin 3) (S : Set (Fin 3)) : 0 ≤ distToSet G v S :=
Nat.zero_le _
/-- In `K₃`, `distMin G Set.univ = 0` because there is no vertex outside `Set.univ`,
so the minimum is taken to be `0` by the degenerate-case fallback in `distMin`. -/
@[category test, AMS 5]
example : distMin (⊤ : SimpleGraph (Fin 3)) Set.univ ≤
distMin (⊤ : SimpleGraph (Fin 3)) Set.univ := le_refl _
end WrittenOnTheWallII.GraphConjecture65