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)/43(n-1)/4) then two copies of G→ are packable into the transitive tournament TTnTTn. 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 32(n-1) then the digraphs G→ and H→ are packable into TTnTTn.