Cops and robbers on directed and undirected abelian Cayley graphs

On the cop number and the weak Meyniel's conjecture for algebraic graphs

“…More recent results on the cop number problem can be found in the works of Schroeder [38], Lu-Peng [32], Scott-Sudakov [39] etc. Recently, Bradshaw [12] and Bradshaw-Hosseini-Turcotte [14] proved the Meyniel's conjecture for abelian Cayley graphs. In fact, there has been a flurry of further activities on the determination of the cop number for classes of graphs.…”

On the cop number of graphs of high girth

“…Lower bounds for cop number are also known for several graph classes. For instance, projective plane incidence graphs with $$2{q}^{2}+2q+2$$ vertices have cop number at least $$q+1$$ [3], and certain abelian Cayley graphs on $$n$$ vertices have cop number $${\rm{\Omega }}(\sqrt{n})$$, with some families achieving their cop number as high as $$\frac{1}{2}\sqrt{n}$$ [7, 9]. Furthermore, Prałat and Wormald [18] showed that random graphs in $${\mathscr{G}}(n,p)$$ with $$pn\gt (\frac{1}{2}+\varepsilon )\mathrm{log}\unicode{x0200A}n$$ and $$\varepsilon \gt 0$$ have cop number of order $$O({n}^{1\unicode{x02215}2})$$ a.a.s.…”

“…For instance, Frankl [8] showed that normal Cayley graphs of degree $$d$$ have cop number at most $$d$$. More recently, it was shown that abelian Cayley graphs on $$n$$ vertices have cop number at most $$0.95\sqrt{n}+\frac{7}{2}$$ [7]. Lower bounds for cop number are also known for several graph classes.…”