Partition of Grids into r-Factors with Components Through Prescribed Vertices

Let [Formula: see text] be the Cartesian product of [Formula: see text] paths, with at least one path having an even number of vertices. Let [Formula: see text] and [Formula: see text] be integers satisfying [Formula: see text] and [Formula: see text], and let [Formula: see text] be a set of [Formula: see text] vertices of [Formula: see text]. We prove that there exists an [Formula: see text]-factor [Formula: see text] of [Formula: see text] consisting of [Formula: see text] components [Formula: see text], where each component [Formula: see text] is an [Formula: see text]-regular, [Formula: see text]-connected, bipancyclic graph containing exactly one vertex of [Formula: see text]. Thus, an [Formula: see text]-dimensional grid of even order can be partitioned into regular, connected and bipancyclic subnetworks passing through prescribed vertices. This extends related results for tori and, as a corollary, also applies to hypercubes.