Meyniel's conjecture on the cop number

Meyniel's conjecture (c. 1985). There is a universal constant C such that every connected graph on n vertices has cop number at most C √n: a team of O(√n) cops can always catch the robber in the game of cops and robbers.

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-meyniel-conjecture,
  title        = {Meyniel's conjecture on the cop number},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/meyniel-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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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

Meyniel's conjecture (c. 1985).

There is a universal constant C such that every connected graph on n vertices has cop number at most C √n: a team of O(√n) cops can always catch the robber in the game of cops and robbers. The conjecture was communicated by Meyniel to Frankl and first appeared in [Fr87]; see [BB12] for a survey.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.MeynielConjecture.

theorem meyniel_conjecture :
    ∃ C : ℝ, 0 < C ∧ ∀ (V : Type) [Fintype V] (G : SimpleGraph V), G.Connected →
      (copNumber G : ℝ) ≤ C * Real.sqrt (Fintype.card V)

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.

Variants

meyniel_conjecture.variants.soft — The soft Meyniel conjecture. There are constants ε, C > 0 such that every connected graph on n vertices has cop number at most C n^(1 - ε).

References

Wikipedia
[Fr87] Frankl, P. (1987). "Cops and robbers in graphs with large girth and Cayley graphs." Discrete Appl. Math. 17, pp. 301--305.
[NW83] Nowakowski, R. and Winkler, P. (1983). "Vertex-to-vertex pursuit in a graph." Discrete Math. 43, pp. 235--239.
[SS11] Scott, A. and Sudakov, B. (2011). "A bound for the cops and robbers problem." SIAM J. Discrete Math. 25, pp. 1438--1442.
[BB12] Baird, W. and Bonato, A. (2012). "Meyniel's conjecture on the cop number: a survey." J. Comb. 3, pp. 225--238.

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.