1. FormalConjectures
  2. WrittenOnTheWallII
  3. GraphConjecture36
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Written on the Wall II - Conjecture 36

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

namespace WrittenOnTheWallII.GraphConjecture36open SimpleGraph

dp G is the number of diametrical pairs of G: the number of unordered pairs {u, v} of vertices at distance diam(G).

noncomputable def dp {α : Type*} [Fintype α] (G : SimpleGraph α) : ℕ := open scoped Classical in (Finset.univ.filter (fun p : Sym2 α => p.lift ⟨fun u v => G.dist u v = G.diam ∧ u ≠ v, fun u v => α:Type u_1inst✝:Fintype αG:SimpleGraph αp:Sym2 αu:αv:α⊢ (fun u v ↦ G.dist u v = G.diam ∧ u ≠ v) u v = (fun u v ↦ G.dist u v = G.diam ∧ u ≠ v) v u All goals completed! 🐙⟩)).card

WOWII Conjecture 36:

For every finite simple connected graph $G$, $\operatorname{path}(G) \ge 2 \cdot \operatorname{rad}(G) / \operatorname{dp}(G)$, where $\operatorname{path}(G)$ is the floor of the average distance of $G$, $\operatorname{rad}(G)$ is the radius of $G$, and $\operatorname{dp}(G)$ is the number of diametrical pairs of $G$ — that is, the number of pairs of vertices at distance $\operatorname{diam}(G)$.

Disproved by Waller in Oct 2003 (counterexample: path number 5, radius 3, dp 1).

@[category research solved, AMS 5] theorem conjecture36 : answer(False) ↔ ∀ {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α], ∀ (G : SimpleGraph α) [DecidableRel G.Adj] (_ : G.Connected) (_ : 0 < dp G), (2 * G.radius.toNat : ℝ) / (dp G : ℝ) ≤ (path G : ℝ) := ⊢ False ↔ ∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj], G.Connected → 0 < dp G → 2 * ↑G.radius.toNat / ↑(dp G) ≤ ↑G.path All goals completed! 🐙-- Sanity checks

The path G invariant is nonneg.

@[category test, AMS 5] example (G : SimpleGraph (Fin 3)) : 0 ≤ (path G : ℝ) := Nat.cast_nonneg _end WrittenOnTheWallII.GraphConjecture36