Packings of graphs and applications to computational complexity

Abstract Let G 1 and G 2 be graphs with n vertices. If there are edge-disjoint copies of G 1 and G 1 with the same n vertices, then we say there is a packing of G 1 and G 2 . This paper is concerned with establishing conditions on G 1 and G 2 under which there is a packing. Our main result (Theorem 1) shows that, with very few exceptions, if G 1 and G 2 together have at most 2 n −3 edges and no vertex is joined to all other vertices, then there is a packing of G 1 and G 2 . Our packing results have some applications to computational complexity. In particular, we show that, for subgraphs of tournaments, the property of containing a sink is a monotone property with minimal computational complexity.