Cop number of random k-uniform hypergraphs

Abstract

The game of \emph{Cops and Robber} is usually played on a graph, in which a group of cops attempt to catch a robber moving along the edges of the graph. The \emph{cop number} of a graph is the minimum number of cops required to win the game. An important conjecture in this area, due to Meyniel, states that the cop number of an $n$-vertex connected graph is $O(\sqrt{n})$. In 2016, Pra\l at and Wormald [Meyniel's conjecture holds for random graphs, Random Structures Algorithms. 48 (2016), no. 2, 396–421. MR3449604] showed that this conjecture holds with high probability for random graphs above the connectedness threshold. Moreoever, \L uczak and Pra\l at [Chasing robbers on random graphs: Zigzag theorem, Random Structures Algorithms. 37 (2010), no. 4, 516–524. MR2760362] showed that on a $\log$-scale the cop number demonstrates a surprising \emph{zigzag} behaviour in dense regimes of the binomial random graph $\Gnp$. In this paper, we consider the game of Cops and Robber on a hypergraph, where the players move along hyperedges instead of edges. We show that with high probability the cop number of the $k$-uniform binomial random hypergraph $G^k(n,p)$ is $O\left(\sqrt{\frac{n}{k}} \log n \right)$ for a broad range of parameters $p$ and $k$. As opposed to the case of $G(n,p)$, on a $\log$-scale our upper bound on the cop number arises as the minimum of \emph{two} complementary zigzag curves. Furthermore, we conjecture that the cop number of a connected $k$-uniform hypergraph on $n$ vertices is $O\left(\sqrt{\frac{n}{k}}\right)$.