# Hunting Rabbits on the Hypercube

## TL;DR

This paper analyzes the Hunters and Rabbits game on hypercubes, providing exact solutions for the hunter number and extending results to scenarios with stationary rabbits, advancing understanding of pursuit-evasion on complex graph structures.

## Contribution

It determines the exact hunter number for hypercubes and generalizes results to cases where the rabbit can stay still between shots.

## Key findings

- Exact hunter number for hypercube $Q^n$ is $1+ extstyle\sum_{i=0}^{n-2} inom{i}{loor{i/2}}$
- Solved the game for graphs with isoperimetric nesting property
- Extended analysis to scenarios with stationary rabbits

## Abstract

We explore the Hunters and Rabbits game on the hypercube. In the process, we find the solution for all classes of graphs with an isoperimetric nesting property and find the exact hunter number of $Q^n$ to be $1+\sum\limits_{i=0}^{n-2} \binom{i}{\lfloor i/2 \rfloor}$. In addition, we extend results to the situation where we allow the rabbit to not move between shots.