Distance two labelling and direct products of graphs Original Research Article

An image-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that vertices at distance two get different numbers and adjacent vertices get numbers which are at least two apart. The image-labelling number of a graph G, image, is the minimum range of labels over all image-labellings of G. Given two graphs G and H, the direct product of G and H is the graph image with vertex set image in which two vertices image and image are adjacent if and only if image and image. In this paper, we completely determine the image-labelling numbers of image for image, and image for image, image, where image is the path of length n. The image-labelling numbers of image for image and some special values of n are also determined, where image is the cycle of length n.