Chip games and paintability

Abstract

We prove that the paint number of the complete bipartite graph KN;N is log N+ O(1). As a consequence, we get that the difference between the paint number and the choice number of KN;N is Ө(log log N). This answers in the negative 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. We obtain that for every on-line two coloring algorithm there exists a k-uniform hypergraph with Ө(2k) edges on which the strategy fails. The results are derived through an analysis of a natural family of chip games.