We show that the cop number of directed and undirected Cayley graphs on abelian groups is in $O\left(\sqrt{n}\right)$, where $n$ is the number of vertices, by introducing a refined inductive method. With our method, we improve the previous upper bound on cop number for undirected Cayley graphs on abelian groups, and we establish an upper bound on the cop number of directed Cayley graphs on abelian groups. We also use Cayley graphs on abelian groups to construct new Meyniel extremal families, which contain graphs of every order $n$ with cop number in $\Theta \left(\sqrt{n}\right)$.

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有向和无向阿贝尔凯莱图上的警察和强盗

我们证明阿贝尔群上有向和无向凯莱图的 cop 数为 $哦\left(\sqrt{n}\right)$， 在哪里 $n$是顶点的数量，通过引入一种改进的归纳方法。使用我们的方法，我们改进了阿贝尔群上无向凯莱图的 cop 数的先前上限，并且我们建立了阿贝尔群上有向凯莱图的 cop 数的上限。我们还使用阿贝尔群上的凯莱图来构造新的梅尼尔极值族，其中包含每个阶的图。$n$ 有警察号码 $\Theta \left(\sqrt{n}\right)$.