FAIR INCOMPLETE TOURNAMENTS WITH ODD NUMBER OF TEAMS
Dalibor Froncek
University of Minnesota Duluth
Suppose n teams are ranked from 1 to n with the strength of the i-th ranked team defined
by s(i) = n + 1 − i. The total strength of opponents that team i plays in a complete round robin
tournament is Sn,n−1 (i) = n(n + 1)/2 − s(i) = (n + 1)(n − 2)/2 + i. A fair incomplete tournament of
n teams with k rounds, FIT(n, k), is a tournament in which every team plays exactly k other teams
and the total strength of the opponents that team i plays is Sn,k (i) = n(n + 1)/2 − s(i) − m for some
fixed constant m. It follows that the total strength of the opponents that each team misses is the
same. An equalized incomplete tournament of n teams with r rounds, EIT(n, r), is a tournament in
which every team plays exactly r other teams and the total strength of the opponents that each team
plays, Tn,r (i), is the same. Obviously, a FIT(n, k) exists if and only if an EIT(n, n − k − 1) exists.
The existence of an EIT(n, r) is equivalent to the existence of an r-regular graph on n vertices with
a 1-vertex-magic vertex labeling with the magic constant m.
It was proved by DF, Tereza Kovarova and Petr Kovar [Bull. ICA 48 (2006), 31–33] that for
n even an EIT(n, r) exists if and only if r is even and either n ≡ 0 (mod 4) or r ≡ 0 (mod 4). We
will present some methods and results for n odd.
Keywords: magic type labeling, round robin tournament, tournament scheduling
email: dalibor@d.umn.edu
Typeset by AMS-TEX