A note on an embedding problem in transitive tournaments

Let T T n be a transitive tournament on n vertices. It is known Görlich, Pilśniak, Woźniak, (2006) [3] that for any acyclic oriented graph G ⃗ of order n and size not greater than 3 4 ( n − 1 ) , two graphs isomorphic to G ⃗ are arc-disjoint subgraphs of T T n . In this paper, we consider the problem of embedding of acyclic oriented graphs into their complements in transitive tournaments. We show that any acyclic oriented graph G ⃗ of size at most 2 3 ( n − 1 ) is embeddable into all its complements in T T n . Moreover, this bound is generally the best possible.