Lower bounds for the game colouring number of partial k-trees and planar graphs

This paper discusses the game colouring number of partial k  -trees and planar graphs. Let colg(PTk)colg(PTk) and colg(P)colg(P) denote the maximum game colouring number of partial k   trees and the maximum game colouring number of planar graphs, respectively. In this paper, we prove that colg(PTk)=3k+2colg(PTk)=3k+2 and colg(P)⩾11colg(P)⩾11. We also prove that the game colouring number colg(G)colg(G) of a graph is a monotone parameter, i.e., if HH is a subgraph of GG, then colg(H)⩽colg(G)colg(H)⩽colg(G).