The game of Cops and Robbers is a well‐known pursuit‐evasion game played on graphs. It has been proved that cubic graphs can have arbitrarily large cop number $c\left(G\right)$, but the known constructions show only that the set $\left\{c\left(G\right)\mid G\text{cubic}\right\}$ is unbounded. In this paper, we prove that there are arbitrarily large subcubic graphs $G$ whose cop number is at least $n^{1∕2-o\left(1\right)}$ where $n=\mid V\left(G\right)\mid$. We also show that proving Meyniel's conjecture for graphs of bounded degree implies a weaker version of Meyniel's conjecture for all graphs.

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有限度图上的梅尼尔猜想

警察与强盗的游戏是一种著名的追逃游戏，使用图表进行游戏。已经证明三次图可以具有任意大的cop数$C（G）$，但已知的构造仅显示该集合 $\left\{C（G）\mid G\text{立方体}\right\}$是无界的。在本文中，我们证明存在任意大的亚三次图$G$ 警察号码至少为 ${ñ}^{\mathrm{1个}∕\mathrm{2个}-Ø（\mathrm{1个}）}$ 在哪里 $ñ=\mid \mathrm{伏特}（G）\mid$。我们还表明，证明有界图的Meyniel猜想意味着对所有图的Meyniel猜想的较弱版本。