Meyniel's conjecture holds for random d ‐regular graphs

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G . The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C , the cop number of every connected graph G is at most C | V ( G ) | . In a separate paper, we showed that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. The result was obtained by showing that the conjecture holds for a general class of graphs with some specific expansion‐type properties. In this paper, this deterministic result is used to show that the conjecture holds asymptotically almost surely for random d ‐regular graphs when d  =  d ( n ) ≥ 3.