Packing of two digraphs into a transitive tournament

Abstract

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 TT n . 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 TT n .