A distance-labelling problem for hypercubes Original Research Article

Let image be integers. An image-labelling of a graph image is a mapping image such that image for any image with image, image, where image is the distance in image between image and image. The integer image is called the label assigned to image under image, and the difference between the largest and the smallest labels is called the span of image. The problem of finding the minimum span, image, over all image-labellings of image arose from channel assignment in cellular communication systems, and the related problem of finding the minimum number of labels used in an image-labelling was originated from recent studies on the scalability of optical networks. In this paper we study the image-labelling problem for hypercubes image (image) and obtain upper and lower bounds on image for any image.