Colouring games on outerplanar graphs and trees

Let f be a graph function which assigns to each graph H a non-negative integer f ( H ) ≤ | V ( H ) | . The f -game chromatic number of a graph G is defined through a two-person game. Let X be a set of colours. Two players, Alice and Bob, take turns colouring the vertices of G with colours from X . A partial colouring c of G is legal (with respect to graph function f ) if for any subgraph H of G , the sum of the number of colours used in H and the number of uncoloured vertices of H is at least f ( H ) . Both Alice and Bob must colour legally (i.e., the partial colouring produced needs to be legal). The game ends if either all the vertices are coloured or there are uncoloured vertices with no legal colour. In the former case, Alice wins the game. In the latter case, Bob wins the game. The f -game chromatic number of G , χ g ( f , G ) , is the least number of colours that the colour set X needs to contain so that Alice has a winning strategy. Let Acy be the graph function defined as Acy ( K 2 ) = 2 , Acy ( C n ) = 3 for any n ≥ 3 and Acy ( H ) = 0 otherwise. Then χ g ( Acy , G ) is called the acyclic game chromatic number of G . In this paper, we prove that any outerplanar graph G has acyclic game chromatic number at most 7. For any integer k , let ϕ k be the graph function defined as ϕ k ( K 2 ) = 2 and ϕ k ( P k ) = 3 ( P k is the path on k vertices) and ϕ k ( H ) = 0 otherwise. This paper proves that if k ≥ 8 then for any tree T , χ g ( ϕ k , T ) ≤ 9 . On the other hand, if k ≤ 6 , then for any integer n , there is a tree T such that χ g ( ϕ k , T ) ≥ n .