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arXiv 2005.10849math.COcs.DM
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关于高围长图的cop数

On the cop number of graphs of high girth

Peter Bradshaw, Seyyed Aliasghar Hosseini, Bojan Mohar, Ladislav Stacho

备注：
18 pages, 1 figure
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摘要：

我们建立了高围长图的cop数的一个下界，它是关于最小度的，更一般地说，是关于一定的增长条件的。特别地，我们证明了具有围长$g$和最小度$δ$的图的copnumber至少为$\tfrac{1}{g}(δ- 1)^{\lfloor \frac{g-1}{4}\rfloor}$ .对于有向图，我们也得到了类似的结果。虽然暴露了几个理由来证明在这个下界中的指数$\tfrac{1}{4}g$不能改进到$(\tfrac{1}{4}+\varepsilon)g$，但我们也能够证明它不能增加到超过$\frac{3}{8}g$。这是通过考虑某个Ramanujan图族而建立的。在我们的证明这个界限，我们还表明，“弱”Meyniel猜想持有有界度的扩展图家庭。

英文摘要：

We establish a lower bound for the cop number of graphs of high girth in terms of the minimum degree, and more generally, in terms of a certain growth condition. We show, in particular, that the cop number of any graph with girth $g$ and minimum degree $δ$ is at least $\tfrac{1}{g}(δ- 1)^{\lfloor \frac{g-1}{4}\rfloor}$. We establish similar results for directed graphs. While exposing several reasons for conjecturing that the exponent $\tfrac{1}{4}g$ in this lower bound cannot be improved to $(\tfrac{1}{4}+\varepsilon)g$, we are also able to prove that it cannot be increased beyond $\frac{3}{8}g$. This is established by considering a certain family of Ramanujan graphs. In our proof of this bound, we also show that the "weak" Meyniel's conjecture holds for expander graph families of bounded degree.

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