Round Robin Tournaments
- From: Maury Barbato <mauriziobarbato@xxxxxxxx>
- Date: Mon, 25 Sep 2006 12:59:53 EDT
Hello,
firt of all, some definitions. In a round robin
tournament each team plays each other team exactly one
time. We have 2n teams. The tournament is organized in
2n-1 rounds: in a round, each team plays exactly one
match. Formally, if we denote by 1,...,2n the teams,
then the i-th round, 1<=i<=2n-1,is the set of all the
matches R_i={{a_1,a_2},...,{a_{2n-1},a_{2n}}},where a_k
is a permutation of {1,...,2n}. A brilliant algorithm to
schedule a round robin tournament is given at the page
http://www.nrich.maths.org/public/viewer.php?obj_id=1443&part=index&refpage=monthindex.php
(join the two lines to obtain the link).
(I)Given a round robin tournament R_i, i=1,...,2n-1, can
you extract a match from each round, such that every
team appear at the most two times in the extracted
matches? I suspect the following fact:
there are two permutations s, t of {1,...,2n-1} such
that, for every 1<=i<2n-1, you can extract the match
{s_i,s_{i+1}} from R_t(i),and the match {s_{2n-1},s_1}
from R_t(2n-1).I have no proof for now.
(II) Suppose that we organize 2 round robin tournaments
with the same 2n teams. We have 2*(2n-1) rounds. Can
you extract a match from each round, such that every
team appear at least one time and at the most four times
in the extracted matches?
Thank you very very much for your ideas.
My Best Regards,
Maury
.
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