Catching a robber on a random k-uniform hypergraph

Abstract

The game of Cops and Robber is usually played on a graph, where a group of cops attempt to catch a robber moving along the edges of the graph. The 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, Prałat and Wormald showed that this conjecture holds with high probability for random graphs above the connectedness threshold. Moreover, Łuczak and Prałat showed that on a $\log $-scale the cop number demonstrates a surprising zigzag behavior in dense regimes of the binomial random graph $G(n,p)$. 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 and that on a $\log $-scale our upper bound on the cop number arises as the minimum of two complementary zigzag curves, as opposed to the case of $G(n,p)$. Furthermore, we conjecture that the cop number of a connected k-uniform hypergraph on n vertices is $O\left (\sqrt {\frac {n}{k}}\,\right )$.