The number of C3-free vertices on 3-partite tournaments

Let T be a 3-partite tournament. We say that a vertex v is −→ C3 -free if v does not lie on any directed triangle of T . Let F3(T ) be the set of the −→ C3 -free vertices in a 3-partite tournament and f3(T ) its cardinality. In this paper we prove that if T is a regular 3-partite tournament, then F3(T )must be contained in one of the partite sets of T . It is also shown that for every regular 3-partite tournament, f3(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 −→ C3 free vertices. Finally we prove that for every c ≥ 3 there exists an infinite family of strongly connected c-partite tournaments, Dc(T ), with n− c − 1 −→ C3 -free vertices. © 2010 Elsevier B.V. All rights reserved.