Clumsy packings of graphs

Let G and H be graphs. We say that P is an H-packing of G if P is a set of edge-disjoint copies of H in G. An H-packing P is maximal if there is no other H-packing of G that properly contains P. Packings of maximum cardinality have been studied intensively, with several recent breakthrough results. Here, we consider minimum cardinality maximal packings. An H-packing P is clumsy if it is maximal of minimum size. Let cl(G,H) be the size of a clumsy H-packing of G. We provide nontrivial bounds for cl(G,H), and in many cases asymptotically determine cl(G,H) for some generic classes of graphs G such as K_n (the complete graph), Q_n (the cube graph), as well as square, triangular, and hexagonal grids. We asymptotically determine cl(K_n,H) for every fixed non-empty graph H. In particular, we prove that where ex(n,H) is the extremal number of H. A related natural parameter is cov(G,H), that is the smallest number of copies of H in G (not necessarily edge-disjoint) whose removal from G results in an H-free graph. While clearly cov(G,H) ≤ cl(G,H), all of our lower bounds for cl(G,H) apply to cov(G,H) as well.