Written on the Wall II - Conjecture 61

WOWII Conjecture 61 For a simple connected graph G, the size f(G) of a largest induced forest satisfies f(G) ≥ residue(G) + ⌈ diam(G) / 3 ⌉, where residue(G) is the Havel-Hakimi residue and diam(G) is the diameter of G.

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-conjecture61,
  title        = {Written on the Wall II - Conjecture 61},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/wall-graph-conjecture61}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

WOWII [Conjecture 61](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)

For a simple connected graph $G$ , the size $f (G)$ of a largest induced forest satisfies $f (G) \geq residue (G) + ⌈ diam (G) /3 ⌉$ , where $residue (G)$ is the Havel-Hakimi residue and $diam (G)$ is the diameter of $G$ .

See: Favaron, Mahéo, Saclé (1991) for the residue; DeLaVina's Graffiti.pc for the conjecture.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.WrittenOnTheWallII.GraphConjecture61.

theorem conjecture61 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
    (residue G : ℝ) + ⌈(G.diam : ℝ) / 3⌉ ≤ (G.largestInducedForestSize : 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.