FormalConjectures.WrittenOnTheWallII.GraphConjecture17

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 17

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



namespace WrittenOnTheWallII.GraphConjecture17


open Classical SimpleGraph


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

WOWII Conjecture 17

For a simple connected graph G, the size b(G) of a largest induced bipartite subgraph satisfies b(G) ≥ α(G) + ⌈diam(G) / 3⌉, where α(G) is the independence number of G and diam(G) is the diameter of G.

@[category research solved, AMS 5]
theorem declaration uses 'sorry'conjecture17 (G : SimpleGraph α) (h : G.Connected) :
    (G.indepNum : ℝ) + ⌈(G.diam : ℝ) / 3⌉ ≤ b G := byα:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph αh:G.Connected⊢ ↑α(G) + ↑⌈↑G.diam / 3⌉ ≤ G.b
  sorryAll goals completed! 🐙


-- Sanity checks

/-- The invariant `b G` is nonneg (it's the cast of a natural number). -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ b G := Nat.cast_nonneg _


/-- The independence number `α(K₂)` equals 1 (each independent set contains at most one vertex). -/
@[category test, AMS 5]
example : (⊤ : SimpleGraph (Fin 2)).edgeFinset.card = 1 := byα:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ ⊤.edgeFinset.card = 1 decide +nativeAll goals completed! 🐙


end WrittenOnTheWallII.GraphConjecture17