Completing partial packings of bipartite graphs

Given a bipartite graph H and an integer n, let f ( n ; H ) be the smallest integer such that any set of edge disjoint copies of H on n vertices can be extended to an H-design on at most n + f ( n ; H ) vertices. We establish tight bounds for the growth of f ( n ; H ) as n → ∞ . In particular, we prove the conjecture of Füredi and Lehel (2010) [4] that f ( n ; H ) = o ( n ) . This settles a long-standing open problem.