Weak Meyniel’s conjecture

Establish that there exists a fixed positive ε such that every connected graph on n vertices has cop number O(n^{1−ε}).

Background

The weak form of Meyniel’s conjecture asks whether the maximal cop number of an n-vertex connected graph is bounded by a polynomial strictly below linear order. The paper states that no positive exponent improvement over the trivial linear-scale bound is currently known, despite the stronger O(√n) conjecture.

This problem is presented as an unresolved weaker target in the general theory of cop numbers, rather than as a question about the limited-visibility model developed later in the paper.

References

Many researchers would consider Meyniel's conjecture to be the main question in the area, but, regarding the state of the art, it is not even known whether there exists a single positive $\varepsilon$ for which the bound $O(n{1-\varepsilon})$ holds (this is sometimes referred to as \emph{weak Meyniel's conjecture}); for the best known upper bounds, an interested reader may check or .

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