FormalConjectures.WrittenOnTheWallII.GraphConjecture142

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 142

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


namespace WrittenOnTheWallII.GraphConjecture142


open Classical SimpleGraph


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

WOWII Conjecture 142:

For a simple connected graph $G$, $\mathrm{tree}(G) \ge (2/3) \cdot \mathrm{girth}(G) + \mathrm{ecc}(B)$ where $\mathrm{tree}(G)$ is the largest induced tree size, $\mathrm{girth}(G)$ is the length of the shortest cycle ($0$ if acyclic), $B$ is the set of boundary vertices (those of maximum eccentricity), and $\mathrm{ecc}(B)$ is the eccentricity of the set $B$.

@[category research open, AMS 5]
theorem declaration uses 'sorry'conjecture142 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
    let B : Set α := (maxEccentricityVertices G : Set α)
    (2 : ℝ) / 3 * (G.girth : ℝ) + (eccSet G B : ℝ) ≤ (largestInducedTreeSize G : ℝ) := byα:Type u_1inst✝³:Fintype αinst✝²:DecidableEq αinst✝¹:Nontrivial αG:SimpleGraph αinst✝:DecidableRel G.Adjh:G.Connected⊢ let B := G.maxEccentricityVertices;
2 / 3 * ↑G.girth + ↑(G.eccSet B) ≤ ↑G.largestInducedTreeSize
  sorryAll goals completed! 🐙


-- Sanity checks

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


/-- `eccSet G` returns a natural number (nonneg). -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) [DecidableRel G.Adj] :
    0 ≤ eccSet G (maxEccentricityVertices G) :=
  Nat.zero_le _


end WrittenOnTheWallII.GraphConjecture142