The game L(d, 1)-labeling problem of graphs

Let G be a graph and let k be a positive integer. Consider the following two-person game which is played on G: Alice and Bob alternate turns. A move consists of selecting an unlabeled vertex upsilon of G and assigning it a number a from {0, 1, 2, . . . , k} satisfying the condition that, for all u epsilon V(G), u is labeled by the number b previously, if d(u, upsilon) = 1, then vertical bar a - b vertical bar >= d, and if d(u, upsilon) = 2, then vertical bar a - b vertical bar >= 1. Alice wins if all the vertices of G are successfully labeled. Bob wins if an impasse is reached before all vertices in the graph are labeled. The game L(d, 1)-labeling number of a graph G is the least k for which Alice has a winning strategy. We use (lambda) over tilde (d)(1)(G) to denote the game L(d, 1)-labeling number of Gin the game Alice plays first, and use (lambda) over tilde (d)(2)(G) to denote the game L(d, 1)-labeling number of G in the game Bob plays first. In this paper, we study the game L(d, 1)-labeling numbers of graphs. We give formulas for (lambda) over tilde (d)(1)(K-n) and (lambda) over tilde (d)(2)(K-n), and give formulas for (lambda) over tilde (d)(1)(K-m,K-n) for those d with d >= max {m, n}. (C) 2012 Elsevier B.V. All rights reserved.