Chip games and paintability

TL;DR

This paper establishes that for complete bipartite graphs, the difference between paint and choice numbers grows logarithmically with the logarithm of N, answering a longstanding question and linking to online hypergraph coloring.

Contribution

It proves the asymptotic difference between paint and choice numbers for $K_{N,N}$ and connects this to online hypergraph coloring strategies.

Findings

Difference between paint and choice numbers is Θ(log log N) for $K_{N,N}$

Existence of hypergraphs with exponential edges where online coloring strategies fail

Analysis of chip games underpins the main results

Abstract

We prove that the difference between the paint number and the choice number of a complete bipartite graph $K_{N,N}$ is $\Theta(\log \log N )$. 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 $\Theta(2^k )$ edges on which the strategy fails. The results are derived through an analysis of a natural family of chip games.