FormalConjectures.WrittenOnTheWallII.GraphConjecture5

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 5

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



namespace WrittenOnTheWallII.GraphConjecture5


open SimpleGraph


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


open Classical

WOWII Conjecture 5

For a simple connected graph G, Ls(G) is bounded below by the maximal size of a sphere of radius radius(G) around the centres of G.

@[category research solved, AMS 5]
theorem declaration uses 'sorry'conjecture5 (G : SimpleGraph V) (h_conn : G.Connected) :
    letI centers := { v : V | G.eccent v = G.radius }
    letI r_nat := G.radius.toNat
    letI sphere_verts (v : V) : Set V := { w | G.dist v w = r_nat }
    letI sphere_size (v : V) : ℝ := ↑(Finset.univ.filter (fun w => w ∈ sphere_verts v)).card
    letI max_sphere_size := sSup (sphere_size '' centers)
    max_sphere_size ≤ Ls G := byV:Type u_1inst✝²:Fintype Vinst✝¹:DecidableEq Vinst✝:Nontrivial VG:SimpleGraph Vh_conn:G.Connected⊢ sSup ((fun v => ↑{w | w ∈ (fun v => {w | G.dist v w = G.radius.toNat}) v}.card) '' {v | G.eccent v = G.radius}) ≤ G.Ls
  sorryAll goals completed! 🐙


end WrittenOnTheWallII.GraphConjecture5