|Department of Mathematics and Statistics|
|University of Minnesota Duluth|
Title: Incomplete round robin tournaments, graph labelings, and magic rectangle sets
Suppose you have a league of n teams that are ranked 1,2,…,n based on their standings in the previous year. If you want them to play a complete round robin tournament, then the last year's winner will apparently have the easiest schedule (they do not play against the strongest team – themselves) and so on, while the weakest team will have the most difficult schedule. What happens when you do not have enough time to play the complete tournament? Maybe you can only schedule g<n-1 games per team. Can you mimic the complete tournament? That is, can you schedule it so that the strongest team has the easiest schedule, while the weakest team has the hardest one? It is easy to see that this is equivalent to scheduling of a tournament with n-1-g games per team, in which the "strength of schedule" (that is, the sum of rankings of the team's opponents) will be the same for each team. We will use graph labelings to show when it is possible to schedule such a tournament. Many of them are based on magic rectangles. However, the most interesting tournament is an incomplete handicap tournament, in which the chances of winning would be the same for each team. In such a tournament the strongest team would have the most difficult schedule, while the weakest team would have the easiest one. We will show such tournaments, whose construction is based on a generalization of magic rectangles, namely magic rectangle sets.