3 research outputs found
Structure theorem for U5-free tournaments
Let be the tournament with vertices , ..., such that , and if , and
. In this paper we describe the tournaments which do not have
as a subtournament. Specifically, we show that if a tournament is
"prime"---that is, if there is no subset , , such that for all , either
for all or for all ---then is
-free if and only if either is a specific tournament or
can be partitioned into sets , , such that , ,
and are transitive. From the prime -free tournaments we can
construct all the -free tournaments. We use the theorem to show that every
-free tournament with vertices has a transitive subtournament with at
least vertices, and that this bound is tight.Comment: 15 pages, 1 figure. Changes from previous version: Added a section;
added the definitions of v, A, and B to the main proof; general edit