In this paper, we propose a problem, called one-to-one disjoint path cover problem, whether or not there exist k disjoint paths joining a pair of vertices which pass through all the vertices other than the two exactly once. A graph which for an arbitrary k, has a one-to-one disjoint path cover between an arbitrary pair of vertices has a hamiltonian property stronger than hamiltonian-connectedness. We investigate this problem on recursive circulants and prove that for an arbitrary k $k(1{leq}k{leq}m)$$ G(2^m,4)$,$m{geq}3$, has a one-to-one disjoint path cover consisting of k paths between an arbitrary pair of vortices.