A distance-labelling problem for hypercubes

Let i"1>=i"2>=i"3>=1 be integers. An L(i"1,i"2,i"3)-labelling of a graph G=(V,E) is a mapping @f:V->{0,1,2,...} such that |@f(u)[email protected](v)|>=i"t for any u,[email protected]?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 @f(v) is called the label assigned to v under @f, and the difference between the largest and the smallest labels is called the span of @f. The problem of finding the minimum span, @l"i"""1","i"""2","i"""3(G), over all L(i"1,i"2,i"3)-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(i"1,i"2,i"3)-labelling was originated from recent studies on the scalability of optical networks. In this paper we study the L(i"1,i"2,i"3)-labelling problem for hypercubes Q"d (d>=3) and obtain upper and lower bounds on @l"i"""1","i"""2","i"""3(Q"d) for any (i"1,i"2,i"3).