The game of cops and robbers is a pursuit game played on a graph G in which a group of cops tries to catch a robber, where both are allowed to move along to edges of G. The cop number of G, denoted by c(G), is the smallest number of cops needed to catch a robber on G. Schröder showed that c(G) is at most 3/2 g(G) + 3, where g(G) is the genus of G, that is, the smallest k such that G can be drawn on an orientable surface of genus k. Furthermore, he conjectured that this bound could be improved to g(G) +3. By relating the game of cops and robbers to a topological game played on a surface we prove that c(G) is at most 4/3 g(G) +3.
|Title of host publication||Proceedings of the European Conference on Combinatorics, Graph Theory and Applications (EuroComb 2019)|
|Publication status||Published - 2019|
|Name||Acta Mathematica Universitatis Comenianae|