FormalConjectures.WrittenOnTheWallII.GraphConjecture33

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 33

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


namespace WrittenOnTheWallII.GraphConjecture33


open Classical SimpleGraph


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

WOWII Conjecture 33

For a simple connected graph G, path(G) ≥ ⌈2 · dist_avg(M, V)⌉, where path(G) is the floor of the average distance of G, M is the set of maximum-degree vertices, and dist_avg(M, V) is the average distance from all vertices to M.

@[category research solved, AMS 5]
theorem declaration uses 'sorry'conjecture33 : answer(False) ↔
    ∀ (α : Type) [Fintype α] [DecidableEq α] [Nontrivial α]
      (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected),
      let M : Set α := {v | G.degree v = G.maxDegree}
      Int.ceil (2 * distavg G M) ≤ (path G : ℤ) := by⊢ False ↔
  ∀ (α : Type) [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α)
    [inst_3 : DecidableRel G.Adj],
    G.Connected →
      let M := {v | G.degree v = G.maxDegree};
      ⌈2 * G.distavg M⌉ ≤ ↑G.path
  sorryAll goals completed! 🐙


-- Sanity checks

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


/-- In `K₃`, the max degree is 2. -/
@[category test, AMS 5]
example : (⊤ : SimpleGraph (Fin 3)).maxDegree = 2 := byα:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ ⊤.maxDegree = 2 decide +nativeAll goals completed! 🐙


end WrittenOnTheWallII.GraphConjecture33