Optimal image-labelings of certain direct products of cycles and Cartesian products of cycles

An image-labeling of a graph image is an assignment of nonnegative integers to the vertices such that adjacent vertices receive labels that differ by at least image and those at a distance of two receive labels that differ by at least one, where image. Let image denote the least image such that image admits an image-labeling using labels from image. We prove that (i) if image, image and image are each a multiple of image, then image, with equality if image, and (ii) if image, image and image are each a multiple of image, then image, with equality if image.