FormalConjectures.WrittenOnTheWallII.GraphConjecture63

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 63

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 $9$, while its largest induced forest and largest induced bipartite subgraph both have order $4$. The conjectured lower bound is therefore $\lceil(9 + 4 + 1)/3\rceil = 5 > 4$.

namespace WrittenOnTheWallII.GraphConjecture63open SimpleGraph

WOWII Conjecture 63 asked whether every simple connected graph $G$ satisfies $f(G) \geq \lceil(\min_v \operatorname{distEven}(v) + b(G) + 1)/3\rceil$. 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/WOWII63.lean#L184-L195"]
theorem conjecture63 : 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! 🐙)
      ⌈((minDistEven : ℝ) + G.b + 1) / 3⌉ ≤ (G.largestInducedForestSize : 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' ⋯;
      ↑⌈(↑minDistEven + G.b + 1) / 3⌉ ≤ ↑G.largestInducedForestSize
  sorryAll goals completed! 🐙

Every vertex is counted among the vertices at even distance from itself.

@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) (v : Fin 3) : 0 < G.distEven v := byG:SimpleGraph (Fin 3)v:Fin 3⊢ 0 < G.distEven v
  unfold SimpleGraph.distEvenG:SimpleGraph (Fin 3)v:Fin 3⊢ 0 < {w | Even (G.dist v w)}.card
  apply Finset.card_pos.mprG:SimpleGraph (Fin 3)v:Fin 3⊢ {w | Even (G.dist v w)}.Nonempty
  exact ⟨v, Finset.mem_filter.mpr
    ⟨Finset.mem_univ _, ⟨0, byG:SimpleGraph (Fin 3)v:Fin 3⊢ G.dist v v = 0 + 0 simp [SimpleGraph.dist_self]All goals completed! 🐙⟩⟩⟩

end WrittenOnTheWallII.GraphConjecture63