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 GG is a forest of maximum degree Δ≥9Δ≥9, then χg(G2)≤colg(G2)≤Δ+3, and there are forests GG with colg(G2)=Δ+3. It is also proved that for an outerplanar graph GG of maximum degree ΔΔ, χg(G2)≤colg(G2)≤2Δ+14, and for a planar graph GG of maximum degree ΔΔ, χg(G2)≤colg(G2)≤23Δ+75.