A distance-labelling problem for hypercubes

Let i1 ≥ i2 ≥ i3 ≥ 1 be integers. An L(i1, i2, i3)-labelling of a graph G = (V, E) is a mapping φ : V → {0, 1, 2, . . .} such that |φ(u)− φ(v)| ≥ it for any u, v ∈ V with d(u, v) = t, t = 1, 2, 3, where d(u, v) is the distance in G between u and v. The integer φ(v) is called the label assigned to v under φ, and the difference between the largest and the smallest labels is called the span of φ. The problem of finding the minimum span, λi1,i2,i3(G), over all L(i1, i2, i3)-labellings of G arose from channel assignment in cellular communication systems, and the related problem of finding the minimum number of labels used in an L(i1, i2, i3)labelling was originated from recent studies of the scalability of optical networks. In this paper we study the L(i1, i2, i3)-labelling problem for hypercubes Qd (d ≥ 3) and obtain upper and lower bounds on λi1,i2,i3(Qd) for any (i1, i2, i3).