Time and space efficient quantum algorithms for detecting cycles and testing bipartiteness

We study space and time-efficient quantum algorithms for two graph problems – deciding whether an n-vertex graph is a forest, and whether it is bipartite. Via a reduction to the s-t connectivity problem, we describe quantum algorithms for deciding both properties in Õ(n^3/2) time whilst using O(log n) classical and quantum bits of storage in the adjacency matrix model. We then present quantum algorithms for deciding the two properties in the adjacency array model, which run in time Õ(n√d_m) and also require O(log n) space, where d_m is the maximum degree of any vertex in the input graph.