FormalConjectures.WrittenOnTheWallII.GraphConjecture85

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 85

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

Counterexample

The conjecture is false for the lexicographic product $C_5[K_4]$. This graph has minimum even-distance count $9$ and largest induced-tree order $4$. The conjectured lower bound is therefore $\lceil\sqrt{1 + 2 \cdot 9}\rceil = \lceil\sqrt{19}\rceil = 5 > 4$.

namespace WrittenOnTheWallII.GraphConjecture85open SimpleGraph

WOWII Conjecture 85 asked whether every simple connected graph $G$ satisfies $\operatorname{tree}(G) \geq \lceil\sqrt{1 + 2\min_v \operatorname{distEven}(v)}\rceil$. The answer is no, as witnessed by $C_5[K_4]$.

@[category research solved, AMS 5, formal_proof using lean4 at
"https://github.com/Kuberwastaken/wowii-63-85-counterexample/blob/cba739842ec59adf7426c180009175b31935701d/lean/WOWII85.lean#L171-L183"]
theorem conjecture85 : answer(False) ↔
    ∀ (α : Type) [Fintype α] [DecidableEq α] [Nontrivial α]
      (G : SimpleGraph α) (_h : G.Connected),
      let minDistEven := (Finset.univ.image (G.distEven ·)).min' (byα:Typeinst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph α_h:G.Connected⊢ (Finset.image (fun x ↦ G.distEven x) Finset.univ).Nonempty simpAll goals completed! 🐙)
      ⌈Real.sqrt (1 + 2 * (minDistEven : ℝ))⌉ ≤
        (G.largestInducedTreeSize : ℝ) := by⊢ False ↔
  ∀ (α : Type) [inst : Fintype α] [DecidableEq α] [inst_2 : Nontrivial α] (G : SimpleGraph α),
    G.Connected →
      let minDistEven := (Finset.image (fun x ↦ G.distEven x) Finset.univ).min' ⋯;
      ↑⌈√(1 + 2 * ↑minDistEven)⌉ ≤ ↑G.largestInducedTreeSize
  sorryAll goals completed! 🐙

The largest induced-tree order is a natural number.

@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ G.largestInducedTreeSize := Nat.zero_le _

end WrittenOnTheWallII.GraphConjecture85