Fast robbers on abelian Cayley graphs and digraphs

Abstract

We study the fast-robber version of the Cops and Robbers game on finite strongly connected abelian Cayley digraphs, including undirected Cayley graphs as the symmetric case. For bounded out-degree $D$, we show that $c_{1,\infty }\left(\Gamma \right)=O_D\left(n^{1-\frac{1}{D}}\right)$; in the undirected case with $D\ge 2$, this improves to the optimal exponent $1-\frac{1}{\lfloor \frac{D}{2}\rfloor }$. We also establish the degree-independent bound $c_{1,\infty }\left(\Gamma \right)=O\left(\frac{n{\left(\log \log n\right)}^2}{\log n}\right)$. These estimates follow from an optimized character-theoretic cyclic sweep over subgroup quotients.