Game colouring of the square of graphs

This paper studies the game chromatic number and game colouring number of the square of graphs. In particular, we prove that if G is a forest of maximum degree Δ ≥ 9 , then χ g ( G 2 ) ≤ col g ( G 2 ) ≤ Δ + 3 , and there are forests G with col g ( G 2 ) = Δ + 3 . It is also proved that for an outerplanar graph G of maximum degree Δ , χ g ( G 2 ) ≤ col g ( G 2 ) ≤ 2 Δ + 14 , and for a planar graph G of maximum degree Δ , χ g ( G 2 ) ≤ col g ( G 2 ) ≤ 23 Δ + 75 .