Extremal problems on consecutive L(2,1)-labelling

For a given graph G of order n, a k-L(2, 1)-labelling is defined as a function f: V(G) -> {0, 1, 2,..., k} such that vertical bar f (u)-f (v)vertical bar >= 2 when d(G) (u, v) = 1 and vertical bar f (u) - f (v)vertical bar >= 1 when d(G) (u, v) = 2. The L(2, 1)-labelling number of G, denoted by lambda(G), is the smallest number k such that G has a k-L(2, I)-labelling. The consecutive L(2, 1)-labelling is a variation of L(2, 1)-labelling under the condition that the labelling f is an onto function. The consecutive L(2, 1)-labelling number of G is denoted by (lambda) over bar (G). Obviously lambda(G) <= (lambda) over bar (G) <= vertical bar V(G)vertical bar - 1 if G admits a consecutive L(2, 1)-labelling. In this paper, we investigate the graphs with (lambda) over bar (G)= vertical bar V(G)vertical bar - 1 and the graphs with (lambda) over bar (G) = lambda(G), in terms of their sizes, diameters and the number of components. (c) 2007 Elsevier B.V. All rights reserved.