The game of cops and robbers is a well-known game played on graphs. In this paper we consider the straight-ahead orientations of 4-regular quadrangulations of the torus and the Klein bottle and we prove that their cop number is bounded by a constant. We also show that the cop number of every k-regularly oriented toroidal grid is at most 13.

中文翻译：

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定向环形网格上的警察和强盗

警察和强盗的游戏是在图表上玩的著名游戏。在本文中，我们考虑了圆环和Klein瓶的4个规则四边形的正方向，并证明了它们的cop数受一个常数限制。我们还显示，每个k规则取向的环形网格的cop数最多为13。