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(√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 Gk(n, p) is O (√ nk log n) 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 (√ nk ).