Lower Bounds for On-line Graph Colorings

We propose three strategies for Presenter in on-line graph coloring games. The first one constructs bipartite graphs and forces any on-line coloring algorithm to use 2 log2 n − 10 colors, where n is the number of vertices in the constructed graph. This is best possible up to an additive constant. The second strategy constructs triangle-free graphs and forces Ω(n 1 2 ) colors. The third one constructs graphs that contain neither C3 nor C5 as a subgraph and forces Ω( n logn 1 3 ) colors. The existence of an on-line algorithm using o(n) colors on triangle-free graphs remains a tantalizing open problem.