Written on the Wall II - Conjecture 189

WOWII Conjecture 189 (status O): For a simple connected graph G, if max_v distEven(v) ≤ 1 + σ(G), then G has a Hamiltonian path. Here distEven(v) is the number of vertices at even distance from v, and σ(G) is the second-smallest degree of G's degree sequence (per WOWII defEntry 65).

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-conjecture189,
  title        = {Written on the Wall II - Conjecture 189},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/wall-graph-conjecture189}},
  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 189](http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj189) (status O):

For a simple connected graph G, if max_v distEven(v) ≤ 1 + σ(G), then G has a Hamiltonian path.

Here distEven(v) is the number of vertices at even distance from v, and σ(G) is the second-smallest degree of G's degree sequence (per WOWII defEntry 65).

Verbatim statement (WOWII #189, status O):

If G is a simple connected graph with n > 1 such that maximum { dist_even(v) : dist_even(v) is the number of vertices at even distance from v } ≤ 1 + σ(G), then G has a Hamiltonian path.

Source: http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj189

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.WrittenOnTheWallII.GraphConjecture189.

theorem conjecture189 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected)
    (hσ : (Finset.univ.image (distEven G)).max' (by simp) ≤ 1 + secondSmallestDegree G) :
    ∃ a b : α, ∃ p : G.Walk a b, p.IsHamiltonian

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).
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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.