Cops and Robbers on graphs of bounded diameter

TL;DR

This paper improves bounds on the number of cops needed to catch a robber in graphs of bounded diameter, using enhanced strategies and probabilistic methods, with specific improvements for diameter three and four graphs.

Contribution

It introduces improved strategies and probabilistic techniques to tighten upper bounds on the cop number for graphs with bounded diameter.

Findings

Upper bound for diameter four reduced to n^{3/5+o(1)}

Upper bound for diameter three reduced to n^{4/7+o(1)}

Enhanced strategies improve previous bounds significantly

Abstract

The game of Cops and Robbers is a well known game played on graphs. In this paper we consider the class of graphs of bounded diameter. We improve the strategy of cops and previously used probabilistic method which results in an improved upper bound for the cop number of graphs of bounded diameter. In particular, for graphs of diameter four, we improve the upper bound from $n^{\frac{2}{3}+o(1)}$ to $n^{\frac{3}{5}+o(1)}$ and for diameter three from $n^{\frac{2}{3}+o(1)}$ to $n^{\frac{4}{7}+o(1)}$.