Cops and robbers on directed and undirected abelian Cayley graphs

Abstract

We discuss the game of cops and robbers on abelian Cayley graphs. We improve the upper bound for cop number in the undirected case, and we give an upper bound for the directed version. We also construct Meyniel extremal families of graphs with cop number Θ( √ n).

On the cop number of graphs of high girth

“…In a similar flavor, Andreae [2] proved that excluding a graph H with t edges as a minor in a graph G implies that c(G) ≤ t. There are also cop number bounds for highly symmetrical graphs. 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 √ n + 2 [7]. Lower bounds for cop number are also known for several graph classes.…”

“…Lower bounds for cop number are also known for several graph classes. For instance, projective plane incidence graphs with 2q 2 + 2q + 2 vertices have cop number at least q + 1 [3], and certain abelian Cayley graphs on n vertices have cop number Ω( √ n), with some families achieving their cop number as high as 1 2 √ n [9,7]. Furthermore, Bollobás et al [4] and Pra lat and Wormald [17] showed that random graphs in G(n, p) have cop number of order Θ( √ n) a.a.s.…”