Written on the Wall II - Conjecture 314

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import FormalConjecturesUtil

/-!
# Written on the Wall II - Conjecture 314

*Reference:*
[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)
-/


namespace WrittenOnTheWallII.GraphConjecture314

open SimpleGraph

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

/-- The size of a largest induced path of $G$, as a natural number.

A subset $s \subseteq V(G)$ is an *induced path* when the induced subgraph
`G.induce s` is a tree in which every vertex has degree at most $2$
(equivalently: a tree that is itself a path graph). We define
`largestInducedPathSize G` as the supremum of `s.card` over all such subsets.

**Disambiguation.** This is *not* the `SimpleGraph.path` invariant, which is the
floor of the average distance — that one is the wrong tool for WOWII Conjecture
314, where $\mathrm{path}(G)$ denotes the **size of a largest induced path**.

TODO: it would probably be clearer to rename the `SimpleGraph.path` invariant to
something like `pathBound` / `floorAvgDist` to avoid this naming collision. -/
noncomputable def largestInducedPathSize (G : SimpleGraph α) [DecidableRel G.Adj] : ℕ :=
  sSup { n | ∃ s : Finset α,
              s.card = n ∧
              (G.induce (s : Set α)).IsTree ∧
              ∀ v : (s : Set α), (G.induce (s : Set α)).degree v ≤ 2 }

/--
WOWII [Conjecture 314](http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj314):

For every finite simple connected graph $G$ with $n > 1$ vertices,
if $G$ is triangle-free and $\mathrm{path}(G) \le 4$, then $G$ is well totally
dominated.

Here $\mathrm{path}(G) = \mathrm{largestInducedPathSize}\, G$ is the **size of a
largest induced path** in $G$, defined locally above.

**Disambiguation.** Earlier revisions of this file used the `SimpleGraph.path`
invariant, but that is the *floor of the average distance*, not the size of a
largest induced path — a different quantity that makes Conjecture 314 vacuous
in many cases.
-/
@[category research open, AMS 5]
theorem conjecture314 [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj]
    (hG : G.Connected)
    (hTriFree : ∀ a b c : α, G.Adj a b → G.Adj b c → G.Adj c a → False)
    (hPath : largestInducedPathSize G ≤ 4) :
    IsWellTotallyDominated G := by
  sorry

-- Sanity checks

/-- The complete graph $K_3$ has a triangle. -/
@[category test, AMS 5]
example : ∃ a b c : Fin 3, (⊤ : SimpleGraph (Fin 3)).Adj a b ∧
    (⊤ : SimpleGraph (Fin 3)).Adj b c ∧ (⊤ : SimpleGraph (Fin 3)).Adj c a := by
  exact ⟨0, 1, 2, by decide, by decide, by decide⟩

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

end WrittenOnTheWallII.GraphConjecture314

• Formalization source ↗

On-site verification needs an executable Lean target. None is attached yet.

A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.