Written on the Wall II - Conjecture 72

WOWII Conjecture 72 For a simple connected graph G, tree(G) ≥ ⌈ecc_avg(G) + max_v λ(v)/3⌉ where tree(G) is the number of vertices in a largest induced tree, ecc_avg(G) is the average eccentricity of G, and max_v l(v) = max_v indepNeighborsCard G v is the maximum over all vertices of the…

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.

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Cite

@misc{cairn-wall-graph-conjecture72,
  title        = {Written on the Wall II - Conjecture 72},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/wall-graph-conjecture72}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

WOWII [Conjecture 72](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)

For a simple connected graph G, $tree (G) \geq ⌈ ecc_{avg} (G) + max_{v} λ (v) /3 ⌉$ where tree(G) is the number of vertices in a largest induced tree, ecc_avg(G) is the average eccentricity of G, and max_v l(v) = max_v indepNeighborsCard G v is the maximum over all vertices of the independence number of the neighbourhood.

Verbatim statement (WOWII #72, status O):

If G is a simple connected graph, then tree(G) ≥ CEIL[average of ecc(v) + maximum of λ(v) /3]

Source: http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj72

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.WrittenOnTheWallII.GraphConjecture72.

theorem conjecture72 (G : SimpleGraph α) (h : G.Connected) :
    let maxL := (Finset.univ.image (fun v => indepNeighborsCard G v)).max' (by simp)
    ⌈averageEccentricity G + (maxL : ℝ) / 3⌉ ≤ (largestInducedTreeSize 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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.