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On-line Algorithms for 2-Coloring Hypergraphs via Chip Games

Summary: On-line Algorithms for 2-Coloring Hypergraphs
via Chip Games
Javed A. Aslam
Laboratory for Computer Science
Massachusetts Institute of Technology
Cambridge, MA 02139
Aditi Dhagat
Department of Electrical Engineering and Computer Science
University of Wisconsin at Milwaukee
Milwaukee, WI 53201
Erd¨os 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. Erd¨os has also shown that
when the number of edges is greater than k22k+1, 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 Lov´asz Local Lemma. In this case, our up-
per bound demonstrates an inherent weakness of on-line strategies by constructing an


Source: Aslam, Javed - College of Computer Science, Northeastern University


Collections: Computer Technologies and Information Sciences