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import FormalConjecturesUtilWritten on the Wall II - Conjecture 316
namespace WrittenOnTheWallII.GraphConjecture316
open SimpleGraph
variable {α : Type*} [Fintype α] [DecidableEq α]
WOWII Conjecture 316
Let G be a simple connected graph and let P denote the set of pendant vertices
(vertices of degree 1). If |P| ≥ deg_avg(Gᶜ), then G is well totally dominated,
where deg_avg(Gᶜ) is the average degree of the complement of G.
@[category research open, AMS 5]
theorem conjecture316 (G : SimpleGraph α) [DecidableRel G.Adj] (hG : G.Connected)
(h : (averageDegree Gᶜ : ℚ) ≤ (pendantVertices G).card) :
IsWellTotallyDominated G := α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αG:SimpleGraph αinst✝:DecidableRel G.AdjhG:G.Connectedh:Gᶜ.averageDegree ≤ ↑G.pendantVertices.card⊢ G.IsWellTotallyDominated
All goals completed! 🐙
-- Sanity checks
/-- The average degree of the edgeless graph on 3 vertices is 0. -/
@[category test, AMS 5]
example : averageDegree (⊥ : SimpleGraph (Fin 3)) = 0 := α:Type u_1inst✝¹:Fintype αinst✝:DecidableEq α⊢ ⊥.averageDegree = 0
α:Type u_1inst✝¹:Fintype αinst✝:DecidableEq α⊢ (∑ v, ↑(⊥.degree v)) / ↑(Fintype.card (Fin 3)) = 0; All goals completed! 🐙
/-- In `P₃` (path 0-1-2), the average degree is 4/3 and there are 2 pendant vertices. -/
@[category test, AMS 5]
example : averageDegree (SimpleGraph.fromEdgeSet {s(0,1), s(1,2)} : SimpleGraph (Fin 3)) = 4/3 := α:Type u_1inst✝¹:Fintype αinst✝:DecidableEq α⊢ (fromEdgeSet {s(0, 1), s(1, 2)}).averageDegree = 4 / 3
α:Type u_1inst✝¹:Fintype αinst✝:DecidableEq α⊢ (∑ v, ↑((fromEdgeSet {s(0, 1), s(1, 2)}).degree v)) / ↑(Fintype.card (Fin 3)) = 4 / 3; All goals completed! 🐙
end WrittenOnTheWallII.GraphConjecture316