FormalConjectures.WrittenOnTheWallII.GraphConjecture144

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 144

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


namespace WrittenOnTheWallII.GraphConjecture144


open Classical SimpleGraph


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

WOWII Conjecture 144

For a simple connected graph $G$, $\mathrm{tree}(G) \ge \mathrm{girth}(G) - 1 + \mathrm{ecc}(\mathrm{Centers})$ where $\mathrm{tree}(G)$ is the largest induced tree size, $\mathrm{girth}(G)$ is the length of the shortest cycle ($0$ if acyclic), $\mathrm{Centers} = G.\mathrm{center}$ is the set of vertices with minimum eccentricity (the center of $G$), and $\mathrm{ecc}(\mathrm{Centers})$ is the eccentricity of the center set — the maximum distance from any non-center vertex to the nearest center vertex.

@[category research open, AMS 5]
theorem declaration uses 'sorry'conjecture144 (G : SimpleGraph α) (h : G.Connected) :
    (G.girth : ℝ) - 1 + (ecc G G.center : ℝ) ≤ (largestInducedTreeSize G : ℝ) := byα:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph αh:G.Connected⊢ ↑G.girth - 1 + ↑(G.ecc G.center) ≤ ↑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 _


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


end WrittenOnTheWallII.GraphConjecture144