CME 305: Discrete Mathematics and Algorithms

Solution: Let G(V,E) denote the graph in question; we construct a new graph H(W,F ) in which an ordinary single particle random walk corresponds to the two particle random walk on G. Take W = V × V = {(v1, v2)|v1, v2 ∈ V } and F = {((v1, v2), (w1, w2))|(v1, w1), (v2, w2) ∈ E}. The idea here is that a uniform random walk on H encodes the state of a two particle random walk on G (in the same way that a random walk on the path graph encodes the state of a drunk). Given a starting configuration v = (v1, v2) ∈ W the expected time until the particles collide is bounded by the hitting time to the vertex u = (v1, v1) ∈ W . This hitting time is bounded by the cover time of H, i.e.