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import FormalConjecturesUtilWritten on the Wall II - Conjecture 20
namespace WrittenOnTheWallII.GraphConjecture20
open Classical SimpleGraph
variable {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α]
WOWII Conjecture 20
For a simple connected graph G, b(G) ≥ n(G) / ⌊deg_avg(G)⌋, where b(G) is
the size of a largest induced bipartite subgraph, n(G) is the number of vertices,
and deg_avg(G) = (∑ v, deg(v)) / n(G) is the average degree.
@[category research solved, AMS 5]
theorem conjecture20 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
let deg_avg : ℝ := (∑ v : α, (G.degree v : ℝ)) / (Fintype.card α : ℝ)
(Fintype.card α : ℝ) / (⌊deg_avg⌋ : ℝ) ≤ (b G : ℝ) := α:Type u_1inst✝³:Fintype αinst✝²:DecidableEq αinst✝¹:Nontrivial αG:SimpleGraph αinst✝:DecidableRel G.Adjh:G.Connected⊢ let deg_avg := (∑ v, ↑(G.degree v)) / ↑(Fintype.card α);
↑(Fintype.card α) / ↑⌊deg_avg⌋ ≤ G.b
All goals completed! 🐙
-- Sanity checks
/-- The number of vertices of `K₃` is 3. -/
@[category test, AMS 5]
example : Fintype.card (Fin 3) = 3 := α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ Fintype.card (Fin 3) = 3 All goals completed! 🐙
/-- The average degree of `K₃` is 2 (every vertex has degree 2 in the complete graph on 3 vertices). -/
@[category test, AMS 5]
example : averageDegree (⊤ : SimpleGraph (Fin 3)) = 2 := α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ ⊤.averageDegree = 2
α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ (∑ v, ↑(⊤.degree v)) / ↑(Fintype.card (Fin 3)) = 2; All goals completed! 🐙
end WrittenOnTheWallII.GraphConjecture20