A d-defective k-painting game on a graph G is played by two players: Lister and Painter. Initially, each vertex is uncolored and has k tokens. In each round, Lister marks a chosen set M of uncolored vertices and removes one token from each marked vertex. In response, Painter colors vertices in a subset X of M which induce a subgraph G[X] of maximum degree at most d. Lister wins the game if at the end of some round there is an uncolored vertex that has no more tokens left. Otherwise, all vertices eventually get colored and Painter wins the game. We say that G is ddefective k-paintable if Painter has a winning strategy in this game. In this paper we show that every planar graph is 3-defective 3-paintable and give a construction of a planar graph that is not 2-defective 3-paintable.

We propose two 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 log 2 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 graphs that contain neither C3 nor C5 as a subgraph and forces Ω( n log n 1 3 ) colors. The best known on-line coloring algorithm for these graphs uses O(n 1 2 ) colors.

(u, v) of G. We study a generalization of the recognition problem where a function ψ defined on a subset V of V (G) is given and the question is whether there is a bar visibility representation ψ of G with ψ(v) = ψ (v) for every v ∈ V . We show that for undirected graphs this problem, and other closely related problems, is NP-complete, but for certain cases involving directed graphs it is solvable in polynomial time.

We study a fractional counterpart of the on-line list colouring game "Mr. Paint and Mrs. Correct" introduced recently by Schauz. We answer positively a question of Zhu by proving that for any given graph the on-line choice ratio and the (off-line) choice ratio coincide. On the other hand it is known from the paper of Alon et al. that the choice ratio equals the fractional chromatic number. It was also shown that the limits used in the definitions of these last two notions can be realised. We show that this is not the case for the on-line choice ratio. Both our results are obtained by exploring the strong links between the on-line choice ratio, and a new on-line game with probabilistic flavour which we introduce.

We prove that the difference between the paint number and the choice number of a complete bipartite graph KN,N is Θ(log log N ). That answers the question of Zhu (2009) whether this difference, for all graphs, can be bounded by a common constant. By a classical correspondence, our result translates to the framework of on-line coloring of uniform hypergraphs. This way we obtain that for every on-line two coloring algorithm there exists a k-uniform hypergraph with Θ(2 k ) edges on which the strategy fails. The results are derived through an analysis of a natural family of chip games.