The game -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 v of G and assigning it a number a from { 0 , 1 , 2 , … , k } satisfying the condition that, for all u ∈ V ( G ) , u is labeled by the number b previously, if d ( u , v ) = 1 , then | a − b | ≥ d , and if d ( u , v ) = 2 , then | a − b | ≥ 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 λ ̃ 1 d ( G ) to denote the game L ( d , 1 ) -labeling number of G in the game Alice plays first, and use λ ̃ 2 d ( 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 λ ̃ 1 d ( K n ) and λ ̃ 2 d ( K n ) , and give formulas for λ ̃ 1 d ( K m , n ) for those d with d ≥ max { m , n } .