Meyniel’s conjecture on the maximal cop number

Prove that the maximal cop number among all connected graphs with n vertices is O(√n) as n tends to infinity, thereby establishing Meyniel’s conjecture.

Background

The paper reviews the classical cop number of a graph, defined as the minimum number of cops required to guarantee capture of a robber. It notes that the asymptotic growth rate of the largest cop number over connected n-vertex graphs is unknown.

Meyniel conjectured in 1985 that this maximum is O(√n). The paper further explains that this bound would be best possible because incidence graphs of finite projective planes have cop number of order √n. Even the weaker assertion that some bound O(n{1−ε}) holds for a fixed positive ε remains unresolved.

References

For example, it is unknown what the growth rate of the maximal cop number among all connected graphs with $n$ vertices (for $n\to\infty$) is. It was conjectured by Meyniel in 1985 to be $O(\sqrt{n})$ (this conjecture was a personal communication to Frankl, as mentioned in in 1987).

— Seeing is not believing in limited visibility cops and robbers  (2507.00941 - Bašić et al., 1 Jul 2025) in Introduction