FormalConjectures.WrittenOnTheWallII.GraphConjecture19

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 19

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



open Classical


namespace WrittenOnTheWallII.GraphConjecture19


open SimpleGraph


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

WOWII Conjecture 19

If G is connected then the size b(G) of a largest induced bipartite subgraph satisfies b(G) ≥ FLOOR((∑ ecc(v))/(|V|) + sSup (range (l G))), where ecc(v) denotes eccentricity and l(G) is the independence number of neighbourhoods.

@[category research open, AMS 5]
theorem declaration uses 'sorry'conjecture19 (G : SimpleGraph α) [Nontrivial α] (h_conn : G.Connected) :
    ⌊(∑ v ∈ Finset.univ, ((G.eccent v).toNat : ℝ)) / (Fintype.card α : ℝ) +
      sSup (Set.range (indepNeighbors G))⌋ ≤ b G := byα:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αG:SimpleGraph αinst✝:Nontrivial αh_conn:G.Connected⊢ ↑⌊(∑ v, ↑(G.eccent v).toNat) / ↑(Fintype.card α) + sSup (Set.range G.indepNeighbors)⌋ ≤ G.b
  sorryAll goals completed! 🐙


end WrittenOnTheWallII.GraphConjecture19