Faculty Research Committee Research Grant
Project: Biclique Cover Number of Upset Tournaments
Abstract of Research Findings
The biclique cover and partition numbers of bipartite graphs and digraphs are related to several matrix ranks. These include the boolean rank, nonnegative integer rank, term rank, and real rank. B.
Shader has characterized singular upset tournament matrices and proved that the nonnegative integer rank of upset tournament matrices is equal to the real rank. A characterization of upset tournaments with respect to their biclique partition number follows directly from Shader’s work. For this research project, we attempted to find such a classification of upset tournaments with respect to their biclique
cover number. If such a characterization is possible, these results, together with Shader's work, will give a complete characterization of upset tournament matrices with respect to each rank and with respect to their biclique cover and partition numbers. So far in our research we have been able to find structures in an upset tournament that contribute to a smaller biclique cover number. However, the proof that these structures must necessarily exist to have a smaller biclique cover number has turned out to be more involved than expected and this research is ongoing. In this final report for this project, we give a proof of a theorem that gives sufficient conditions for an upset tournament to have a given upper bound for the biclique cover number and we give a best possible lower bound for the biclique cover number in terms of the size of the tournament. Lastly, we give a conjecture and partial proof that certain structures necessarily occur a specific number of times in an upset tournament for a given biclique cover number. If this conjecture can be proven, this will give a characterization of upset tournaments with respect to their biclique cover number, as desired.
Submitted by:
Daluss Siewert
Associate Professor of Mathematics
January 18, 2010