L(2,1)-Labelings on the composition of n graphs

An L(2, 1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that vertical bar f (x) f (y)vertical bar >= 2 if d(x, y) = 1 and vertical bar f (x) f (y)vertical bar >= 1 if d(x, y) = 2, where d(x, y) denotes the distance between x and y in G. The L(2, 1)-labeling number lambda(G) of G is the smallest number k such that G has an L(2, 1)-labeling with max{f (v) : v is an element of V(G)} = k. Griggs and Yeh conjectured that lambda(G) <= Delta(2) for any simple graph with maximum degree Delta >= 2. In this article, a problem in the proof of a theorem in Shao and Yeh (2005)[19] is addressed and the graph formed by the composition of n graphs is studied. We obtain bounds for the L(2, 1)-labeling number for graphs of this type that is much better than what Griggs and Yeh conjectured for general graphs. As a corollary, the present bound is better than the result of Shiu et al. (2008) [21] for the composition of two graphs G(1) [G(2)] if v(2) < Delta(2)(2)+ 1, where v(2) and Delta(2) are the number of vertices and maximum degree of G(2) respectively. (C) 2010 Elsevier B.V. All rights reserved.