Question 81
Let $G$ be a graph of diameter $2$ on $2019$ vertices. $62$ cops and $1$ robber are placed on the vertices of graph $G$. Everybody sees all others, they move in turn. Each move each person can stay in his vertex or go to any of the adjacent vertices (each cop stay or go independently of the others and they coordinate their actions during a game). The robber makes the first move, then the cops move (simultaneously), then the robber moves again, then cops, etc. The robber is caught if he is in the vertex occupied by cop. Prove that the cops can catch the robber.