Written on the Wall II - Conjecture 84

WOWII Conjecture 84 (status O): For a simple connected graph G with positive minimum degree, tree(G) ≥ 2 · rad(G) / δ(G), where tree(G) is the number of vertices in a largest induced subtree, rad(G) is the radius, and δ(G) = G.minDegree is the minimum degree.

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-conjecture84,
  title        = {Written on the Wall II - Conjecture 84},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/wall-graph-conjecture84}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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The problem

The question

WOWII [Conjecture 84](http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj84) (status O):

For a simple connected graph G with positive minimum degree, tree(G) ≥ 2 · rad(G) / δ(G), where tree(G) is the number of vertices in a largest induced subtree, rad(G) is the radius, and δ(G) = G.minDegree is the minimum degree.

The hypothesis 0 < G.minDegree follows from connectedness on a nontrivial vertex set, but is stated explicitly so the theorem reads as a real-valued inequality without x / 0 corner cases.

Verbatim statement (WOWII #84, status O):

If G is a simple connected graph, then tree(G) ≥ 2 rad(G) / δ(G)

Source: http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj84

Here δ(G) is the minimum degree of G (lowercase delta in the WOWII HTML, rendered as <font face="Symbol">d</font>), not the average distance.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.WrittenOnTheWallII.GraphConjecture84.

theorem conjecture84 (G : SimpleGraph α) [DecidableRel G.Adj]
    (h : G.Connected) (hδ : 0 < G.minDegree) :
    2 * (G.radius.toNat : ℝ) / (G.minDegree : ℝ) ≤ (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.
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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.