On the existence of sports schedules with multiple venuesAlgebra & Discrete Mathematics
|Speaker:||Yoshiko Ikebe, Tokyo University of Science|
|Start time:||Wed, Mar 12 2008, 3:10PM|
We consider the problem of scheduling a sports event for 2n teams and n stadiums which have no association with the various teams. We wish to create a schedule satsifying the conditions: (1) all team pairs play against each other at least once, (2) n games are simultaneously played each day, (3) each team plays exactly twice in each stadium, and (4) no two teams play against each other twice in the same stadium. Since each teams plays 2n games, the resulting schedule consists of the usual round robin with one additional round, and for each team there is exactly one partner whom it meets twice. de Werra, Ekim and Raess showed that such a schedule always exists when 2n is a power of 2 greater than or equal to 8. In this talk we will give a constructive proof that there always exists such a schedule for any 2n greater than or equal to 10. Various tools such as Kirkman's circle method and primitive roots of prime numbers will be used.