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

Abstract Erdos has shown that, for all k -hypergraphs with fewer than 2 k −1 edges, there exists a 2-coloring of the nodes so that no edge is monochromatic. Erdos has also shown that, when the number of edges is greater than k 2 2 k +1 , there exist k -hypergraphs with no such 2-coloring. These bounds are not constructive, however. In this paper, we taken 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 Lovasz 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 -hypergraphs with (3 + 2 √2) k edges.