The number of -free vertices on 3-partite tournaments

Let T be a 3-partite tournament. We say that a vertex v is C 3 ⃗ -free if v does not lie on any directed triangle of T . Let F 3 ( T ) be the set of the C 3 ⃗ -free vertices in a 3-partite tournament and f 3 ( T ) its cardinality. In this paper we prove that if T is a regular 3-partite tournament, then F 3 ( T ) must be contained in one of the partite sets of T . It is also shown that for every regular 3-partite tournament, f 3 ( T ) does not exceed n 9 , where n is the order of T . On the other hand, we give an infinite family of strongly connected tournaments having n − 4 C 3 ⃗ -free vertices. Finally we prove that for every c ≥ 3 there exists an infinite family of strongly connected c -partite tournaments, D c ( T ) , with n − c − 1 C 3 ⃗ -free vertices.