Packing of two digraphs into a transitive tournament

Let −→ G and −→ H be two oriented graphs of order n without directed cycles. Görlich, Pilśniak and Woźniak proved [A note on a packing problem in transitive tournaments, preprint Faculty of Applied Mathematics, AGH University of Science and Technology, No. 37/2002] that if the number of arcs in −→ G is sufficiently small (not greater than 3(n − 1)/4) then two copies of −→G are packable into the transitive tournament TTn. This bound is best possible. In this paper we give a generalization of this result. We show that if the sum of sizes of −→ G and −→ H is not greater than 3 2 (n − 1) then the digraphs −→ G and −→ H are packable into TTn. © 2006 Elsevier B.V. All rights reserved.