Catching a robber on a random $k$-uniform hypergraph

TL;DR

This paper extends the Cops and Robber game to hypergraphs, providing probabilistic bounds on the cop number for random hypergraphs and revealing a zigzag pattern in its behavior.

Contribution

It introduces the study of Cops and Robber on hypergraphs, establishing upper bounds on the cop number for random hypergraphs and identifying a new zigzag pattern in its scale.

Findings

Cop number of random hypergraphs is O(√(n/k) log n) for broad parameters.

The upper bound exhibits a zigzag pattern on a log scale, with two complementary curves.

Conjecture: cop number of connected hypergraphs is O(√(n/k)).

Abstract

The game of \emph{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 \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 $G(n,p)$. 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 \emph{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 $O\left(\sqrt{\frac{n}{k}}\,\right)$.