AI 中文总结
该研究针对有限强连通阿贝尔凯莱有向图的警察与强盗游戏快速强盗版本，推导了有界出度下的复杂度上界及缓增度区域的一致次线性界，改进了无向情形的结果。
AI 中文摘要
我们研究有限强连通阿贝尔凯莱有向图上的警察与强盗游戏的快速强盗版本，其中无向凯莱图为对称情形。对有界出度D，我们证明c₁,∞(Γ)=O_D(n^{1-1/D})，无向情形下改进为O_D(n^{1-2/D})，还在更广泛的缓增度区域中证明了一致次线性界。
英文摘要
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 $c_{1,\infty}\left(Γ\right)=O_D\left(n^{1-\frac{1}{D}}\right)$; in the undirected case with $D\ge2$, this improves to the optimal exponent $1-\frac{1}{\left\lfloor \frac{D}{2}\right\rfloor}$. We also establish the degree-independent bound $c_{1,\infty}\left(Γ\right)=O\!\left(\frac{n\left(\log\log n\right)^2}{\log n}\right)$. These estimates follow from an optimized character-theoretic cyclic sweep over subgroup quotients.
