Cops and robbers on directed and undirected abelian Cayley graphs

@article{Bradshaw2021CopsAR,
  title={Cops and robbers on directed and undirected abelian Cayley graphs},
  author={Peter Bradshaw and Seyyed Aliasghar Hosseini and J'er'emie Turcotte},
  journal={Eur. J. Comb.},
  year={2021},
  volume={97},
  pages={103383}
}
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 $\Theta (\sqrt{n})$. 
1 Citations

Figures from this paper

On the cop number of graphs of high girth
TLDR
It is shown that the "weak" Meyniel's conjecture holds for expander graph families of bounded degree and that the cop number of any graph with girth $g$ and minimum degree $\delta$ is at least $\tfrac{1}{g}(\delta - 1)^{\lfloor g-1}{4}\rfloor}$.

References

SHOWING 1-10 OF 35 REFERENCES
On a Pursuit Game on Cayley Digraphs
TLDR
It is proved in the Abelian undirected case that 3 k 4 Ȱ pursuers can catch the evader, where k is the degree.
On a pursuit game on
  • Cayley graphs. Combinatorica,
  • 1987
Meyniel’s conjecture on the cop number: A survey
Meyniel’s conjecture is one of the deepest open problems on the cop number of a graph. It states that for a connected graph G of order n, c(G) = O( √ n). While largely ignored for over 20 years, the
Sidon Sets in Groups and Induced Subgraphs of Cayley Graphs
TLDR
It is proved that most graphs on n vertices are not induced subgraphs of any vertex transitive graph with the help of Fourier analysis.
A proof of the Meyniel conjecture for Abelian Cayley graphs
Abstract We prove that the cop number of a connected abelian Cayley graph on n vertices is bounded by 7 n . This proves that H. Meyniel’s conjectured bound of O ( n ) for the cop number of any
Cops and robbers on oriented toroidal grids
TLDR
This paper considers the straight-ahead orientations of 4-regular quadrangulations of the torus and the Klein bottle and proves that their cop number is bounded by a constant.
Kozma's. Useful inequalities
  • 2020
Useful inequalities. Available at http:// www. lkozma.net/ inequalities_cheat_ sheet/ ineq
  • 2020
Version 12.1. Champaign, IL
  • 2020
A proof of the Meyniel conjecture for abelian Cayley graphs
  • Discrete Mathematics,
  • 2019
...
1
2
3
4
...