On-Line Algorithms for 2-Coloring Hypergraphs Via Chip Games

Erdös has shown that for all k-hypergraphs with fewer than 2k−1 edges, there exists a 2-coloring of the nodes so that no edge is monochromatic. Erdös has also shown that when the number of edges is greater than k2, there exist k-hypergraphs with no such 2-coloring. These bounds are not constructive, however. In this paper, we take an “on-line” look at this problem, showing constructive upper and lower bounds on the number of edges of a hypergraph which allow it to be 2-colored on-line. These bounds become particularly interesting for degree-k k-hypergraphs which always have a good 2-coloring for all k ≥ 10 by the Lovász Local Lemma. In this case, our upper bound demonstrates an inherent weakness of on-line strategies by constructing an adversary which defeats any on-line 2-coloring algorithm using degree-k k-hypergraph with (3 + 2 √ 2) edges.