A distance-labelling problem for hypercubes

Abstract

Let i1 ≥ i2 ≥ i3 ≥ 1 be integers. An L (i1, i2, i3)-labelling of a graph G = (V, E) is a mapping φ{symbol} : V → {0, 1, 2, ...} such that | φ{symbol} (u) - φ{symbol} (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{symbol} (v) is called the label assigned to v under φ{symbol}, and the difference between the largest and the smallest labels is called the span of φ{symbol}. 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 on 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). © 2007 Elsevier B.V. All rights reserved.