Source labels openChecked July 26, 2026

Written on the Wall II - Conjecture 314

Mathematical statement

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.

Statement source: Written on the Wall II statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

conjecture314

Canonical source

Complete statement target, proof intentionally absentLean 4

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

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization Conjecture 314background E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pcbackground