Written on the Wall II - Conjecture 133

WOWII Conjecture 133: For a simple connected graph G, path(G) ≥ rad(G) + (avg_v l(v))^cC_4(G), where path(G) is the path number of the graph (number of vertices of a largest induced path), rad(G) is the radius (minimum eccentricity, as a natural number), avg_v l(v) = l(G) is the average…

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

Start working on it Submit a claim Follow

Cite

@misc{cairn-wall-graph-conjecture133,
  title        = {Written on the Wall II - Conjecture 133},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/wall-graph-conjecture133}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

WOWII [Conjecture 133](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/):

For a simple connected graph $G$ , $\operatorname{path}(G) \ge \operatorname{rad}(G) + (\mathrm{avg}_v\, l(v))^{cC_4(G)}$, where $path (G)$ is the path number of the graph (number of vertices of a largest induced path), $rad (G)$ is the radius (minimum eccentricity, as a natural number), $avg_{v} l (v) = l (G)$ is the average independence number of vertex neighbourhoods, and $c C_{4} (G)$ is the $C_{4}$ -free characteristic function (1 if $G$ is $C_{4}$ -free, not necessarily induced, and 0 otherwise).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.WrittenOnTheWallII.GraphConjecture133.

theorem conjecture133 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
    let rad := G.radius.toNat
    let hasC4 := ∃ a b c d : α, a ≠ b ∧ a ≠ c ∧ a ≠ d ∧ b ≠ c ∧ b ≠ d ∧ c ≠ d ∧
      G.Adj a b ∧ G.Adj b c ∧ G.Adj c d ∧ G.Adj d a
    let cC4 : ℕ := if hasC4 then 0 else 1
    (rad : ℝ) + l G ^ cC4 ≤ (path G : ℝ)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

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

Source and licence

Imported from Formal Conjectures (Written on the Wall II), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.