## Abstract

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.

Original language | English |
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Article number | P2.39 |

Journal | Electronic Journal of Combinatorics |

Volume | 26 |

Issue number | 2 |

DOIs | |

State | Published - 2019 |

### Bibliographical note

Publisher Copyright:© The authors.

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics