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import FormalConjecturesUtilWritten on the Wall II - Conjecture 194
namespace WrittenOnTheWallII.GraphConjecture194
open Classical SimpleGraph
variable {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α]
WOWII Conjecture 194
For a simple connected graph G, if α(G) ≤ 1 + l_avg(G), then G has a Hamiltonian path.
Here α(G) = G.indepNum is the independence number, and
l_avg(G) = averageIndepNeighbors G is the average over all vertices of the independence number
of the neighbourhood.
A Hamiltonian path is a walk visiting every vertex exactly once.
@[category research open, AMS 5]
theorem conjecture194 (G : SimpleGraph α) (h : G.Connected)
(hα : (G.indepNum : ℝ) ≤ 1 + averageIndepNeighbors G) :
∃ a b : α, ∃ p : G.Walk a b, p.IsHamiltonian := α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph αh:G.Connectedhα:↑α(G) ≤ 1 + G.averageIndepNeighbors⊢ ∃ a b p, p.IsHamiltonian
All goals completed! 🐙
-- Sanity checks
/-- The average indep-neighbors invariant `l G` is nonneg. -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ averageIndepNeighbors G := α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph (Fin 3)⊢ 0 ≤ G.averageIndepNeighbors
α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph (Fin 3)⊢ 0 ≤ ∑ v, G.indepNeighbors vα:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph (Fin 3)⊢ 0 ≤ ↑(Fintype.card (Fin 3))
α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph (Fin 3)⊢ 0 ≤ ∑ v, G.indepNeighbors v α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph (Fin 3)⊢ ∀ i ∈ Finset.univ, 0 ≤ G.indepNeighbors i
intro v α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph (Fin 3)v:Fin 3a✝:v ∈ Finset.univ⊢ 0 ≤ G.indepNeighbors v
All goals completed! 🐙
α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph (Fin 3)⊢ 0 ≤ ↑(Fintype.card (Fin 3)) All goals completed! 🐙
/-- The edgeless graph on 2 vertices has 2 vertices. -/
@[category test, AMS 5]
example : Fintype.card (Fin 2) = 2 := α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ Fintype.card (Fin 2) = 2 All goals completed! 🐙
end WrittenOnTheWallII.GraphConjecture194