Short Score Certificates for Upset Tournaments

A score certificate for a tournament, T , is a collection of arcs of T which can be uniquely completed to a tournament with the same scorelist as T ’s, and the score certificate number of T is the least number of arcs in a score certificate of T . Upper bounds on the score certificate number of upset tournaments are derived. The upset tournaments on n vertices are in one–to–one correspondence with the ordered partitions of n−3, and are “almost” transitive tournaments. For each upset tournament on n vertices a general construction of a score certificate with at most 2n− 3 arcs is given. Also, for the upset tournament, Tλ, corresponding to the ordered partition λ, a score certificate with at most n+ 2k+ 3 arcs is constructed, where k is the number of parts of λ of size at least 2. Lower bounds on the score certificate number of Tλ in the case that each part is sufficiently large are derived. In particular, the score certificate number of the so-called nearly transitive tournament on n vertices is shown to be n+ 3, for n ≥ 10.