Labelling Graphs with the Circular Difference∗

For positive integers k and d ≥ 2, a k-S(d, 1)-labelling of a graph G is a function on the vertex set of G, f : V (G) → {0, 1, 2, · · · , k− 1}, such that |f(u)− f(v)|k ≥ { d if dG(u, v) = 1; 1 if dG(u, v) = 2, where |x|k = min{|x|, k − |x|} is the circular difference modulo k. In general, this kind of labelling is called the S(d, 1)-labelling. The σdnumber of G, σd(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 kS′(d, 1)-labelling, S′(d, 1)-labelling and σ′ d(G). If the circular difference 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 difference of the maximum and the minimum labels used. The λd-number of G, λd(G), is defined as the minimum span among all L(d, 1)-labellings of G. In this case, we have the corresponding L′(d, 1)-labelling and λd(G) for the labelling with injective condition. We will first study the relation between λd and σd as well λd and σ′ d. Then we consider these parameters on cycles and trees. Finally, we study the join of graphs and the multipartite graphs.