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Level A · Machine-checkable Hard Graph theory P-wall-graph-conjecture96

Written on the Wall II - Conjecture 96

WOWII Conjecture 96 For a simple connected graph G on at least two vertices, α(G) ≤ 1 + min_v distEven(v) · (max_v l(v) - 1) where α(G) = G.indepNum is the independence number, distEven(v) is the number of vertices at even distance from v, and max_v l(v) is the maximum independence number of a…

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-conjecture96,
  title        = {Written on the Wall II - Conjecture 96},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/wall-graph-conjecture96}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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or , ,

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The problem

The question

WOWII [Conjecture 96](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)

For a simple connected graph G on at least two vertices, α(G) ≤ 1 + min_v distEven(v) · (max_v l(v) - 1) where α(G) = G.indepNum is the independence number, distEven(v) is the number of vertices at even distance from v, and max_v l(v) is the maximum independence number of a vertex neighbourhood.

We record the nontrivial-graph version: the literal source inequality fails on , where the independence number and even-distance count are both one, but the maximum local independence number is zero, giving .

Verbatim statement (WOWII #96, status O):

If G is a simple connected graph, then α(G) ≤ 1 + minimum dist_even(v)*(maximum of λ(v) -1)

Source: http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj96

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.WrittenOnTheWallII.GraphConjecture96.

theorem conjecture96 (G : SimpleGraph α) (h : G.Connected) :
    let minDistEven := (Finset.univ.image (distEven G)).min' (by simp)
    let maxL := (Finset.univ.image (fun v => indepNeighborsCard G v)).max' (by simp)
    (G.indepNum : ℝ) ≤ 1 + (minDistEven : ℝ) * ((maxL : ℝ) - 1)

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.