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arXiv · 2005.10849

On the cop number of graphs of high girth

Peter Bradshaw·Seyyed Aliasghar Hosseini·Bojan Mohar·Ladislav Stacho
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Abstract

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 $\delta$ is at least $\tfrac{1}{g}(\delta - 1)^{\lfloor \frac{g-1}{4}\rfloor}$. 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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Peter Bradshaw, Seyyed Aliasghar Hosseini, Bojan Mohar, Ladislav Stacho. 2020-05-21. On the cop number of graphs of high girth. https://arxiv.org/abs/2005.10849

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