Meyniel's conjecture on graphs of bounded degree

The game of Cops and Robbers is a well‐known pursuit‐evasion game played on graphs. It has been proved that cubic graphs can have arbitrarily large cop number c ( G ), but the known constructions show only that the set { c ( G ) ∣ G cubic } is unbounded. In this paper, we prove that there are arbitrarily large subcubic graphs G whose cop number is at least n 1 ∕ 2 − o ( 1 ) where n = ∣ V ( G ) ∣. We also show that proving Meyniel's conjecture for graphs of bounded degree implies a weaker version of Meyniel's conjecture for all graphs.