L(p, 2, 1)-labeling of the infinite regular trees

An L(p, q, r)-labeling of a graph G is defined as a function f from the vertex set V(G) into the nonnegative integers such that for any two vertices x, y, [f(x) - f(y)vertical bar >= p if d(x, y) = 1, vertical bar f(x) - f(y)vertical bar >= q if d(x, y) = 2 and vertical bar f(x) - f(y)vertical bar >= r if d(x, y) = 3, where d(x, y) is the distance between x and y in G. The L(p, q, r)-labeling number of G is the smallest number k such that G has an L(p, q, r)-labeling with k = max{f(x): x is an element of V(G)}. In this paper, we obtain all the L(p, 2, 1)-labeling numbers of the infinite D-regular trees T-infinity(D) for p >= 2 and D >= 3. In all cases, we also construct an optimal L(p, 2, 1)-labeling of T-infinity(D). (C) 2013 Elsevier B.V. All rights reserved.