FormalConjectures.WrittenOnTheWallII.GraphConjecture34

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 34

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


namespace WrittenOnTheWallII.GraphConjecture34


open Classical SimpleGraph


variable {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α]

WOWII Conjecture 34

For a simple connected graph G, path(G) ≥ ⌈dist_avg(C, V) + dist_avg(M, V)⌉, where path(G) is the floor of the average distance of G, C is the set of center vertices (those with minimum eccentricity), M is the set of maximum-degree vertices, and dist_avg(S, V) is the average distance from all vertices to the set S.

@[category research solved, AMS 5]
theorem declaration uses 'sorry'conjecture34 :
  answer(sorry) ↔
    ∀ (α : Type) [Fintype α] [DecidableEq α] [Nontrivial α]
      (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected),
      let C : Set α := center G
      let M : Set α := {v | G.degree v = G.maxDegree}
      Int.ceil (distavg G C + distavg G M) ≤ (path G : ℤ) := by⊢ True ↔
  ∀ (α : Type) [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α)
    [inst_3 : DecidableRel G.Adj],
    G.Connected →
      let C := G.center;
      let M := {v | G.degree v = G.maxDegree};
      ⌈G.distavg C + G.distavg M⌉ ≤ ↑G.path
  sorryAll goals completed! 🐙


-- Sanity checks

/-- The `path G` invariant is nonneg when cast to ℤ. -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ (path G : ℤ) := Int.natCast_nonneg _


/-- The edgeless graph on 3 vertices has no edges. -/
@[category test, AMS 5]
example : (⊥ : SimpleGraph (Fin 3)).edgeFinset.card = 0 := byα:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ ⊥.edgeFinset.card = 0 decide +nativeAll goals completed! 🐙


end WrittenOnTheWallII.GraphConjecture34