Labelling graphs with the circular difference

For positive integers k and d greater than or equal to 2, a k-S(d, I.)-labe lling of a graph G is a function on the vertex set of G, f : V(G) --> {0, 1 , 2,...,k-1}, such that [GRAPHICS] where \x\(k) = min{\x\, k-\x\} is the circular difference module k. In gene ral, this kind of labelling is called the S(d, 1)-labelling. The sigma (d)- number of G, sigma (d)(G), is the minimum k of a k-S(d, 1)-labelling of G. If the labelling is required to be injective, then we have analogous k-S'(d , 1)-labelling, S' (d, 1.)labelling and sigma (d)'(G). If the circular diff erence in the definition above is replaced by the absolute difference, then f is an L(d, 1)-labelling of G. The span of an L(d, 1)-labelling is the di fference of the maximum and the minimum labels used. The lambda (d)-number of G, lambda (d)(G), is defined as the minimum span among all L(d, 1)labell ings of G. In this case, we have the corresponding L'(d, 1)-labelling and l ambda (d)'(G) for the labelling with injective condition. We will first study the relation between lambda (d) and sigma (d) as well l ambda (d)' and sigma (d)'. Then we consider these parameters on cycles and trees. Finally we study the join of graphs and the multipartite graphs.