FormalConjectures.WrittenOnTheWallII.GraphConjecture103

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 103

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


namespace WrittenOnTheWallII.GraphConjecture103


open Classical SimpleGraph


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

WOWII Conjecture 103

For a simple connected graph $G$, $\alpha(G) \le \lfloor b(G) - \ln(\mathrm{ecc_avg}(G)) \rfloor$ where $\alpha(G) = G.\mathrm{indepNum}$ is the independence number, $b(G)$ is the largest induced bipartite subgraph size, and $\mathrm{ecc_avg}(G) = G.\mathrm{averageEccentricity}$ is the average eccentricity of $G$. Uses Real.log (natural logarithm).

@[category research open, AMS 5]
theorem declaration uses 'sorry'conjecture103 (G : SimpleGraph α) (h : G.Connected) :
    (G.indepNum : ℝ) ≤ ⌊b G - Real.log (averageEccentricity G)⌋ := byα:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph αh:G.Connected⊢ ↑α(G) ≤ ↑⌊G.b - Real.log G.averageEccentricity⌋
  sorryAll goals completed! 🐙


-- Sanity checks

/-- The independence number is nonneg. -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ G.indepNum := Nat.zero_le _


/-- `Real.log` of a positive real is well-defined. -/
@[category test, AMS 5]
example : Real.log 1 = 0 := Real.log_one


end WrittenOnTheWallII.GraphConjecture103