On <i>r</i> -factors with components containing specified vertices in a torus

Let T be a graph obtained by taking the Cartesian product of n cycles of even lengths [Formula: see text]. It is known that for any subset of k vertices of T with [Formula: see text] there exists a 2-factor in T where each cycle contains exactly one of the k vertices. We generalize this result to r-factors by proving that for [Formula: see text] and [Formula: see text] there exists an r-factor H in T such that each component of H is an r-regular, r-connected and bipancyclic graph, with each component containing exactly one of the k specified vertices of T. As a consequence, we obtain an analogous result for hypercubes.