### Abstract

Let d be a positive integer. A graph G is called d-divisible if d divides the degree of each vertex of G. G is called nowhere d-divisible if no degree of a vertex of G is divisible by d. For a graph H, gcd(H) denotes the greatest common divisor of the degrees of the vertices of H. The H-packing number of G is the maximum number of pairwise edge disjoint copies of H in G. The H-covering number of G is the minimum number of copies of H in G whose union covers all edges of G. Our main result is the following: For every fixed graph H with gcd(H) = d, there exist positive constants ∈(H) and N(H) such that if G is a graph with at least N(H) vertices and has minimum degree at least (1 - ∈(H))|G|, then the H-packing number of G and the H-covering number of G can be computed in polynomial time. Furthermore, if G is either d-divisible or nowhere d-divisible, then there is a closed formula for the H-packing number of G, and the H-covering number of G. Further extensions and solutions to related problems are also given.

Original language | English (US) |
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Pages (from-to) | 451-472 |

Number of pages | 22 |

Journal | Journal of Combinatorial Designs |

Volume | 6 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 1998 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

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## Cite this

*Journal of Combinatorial Designs*,

*6*(6), 451-472. https://doi.org/10.1002/(SICI)1520-6610(1998)6:6<451::AID-JCD6>3.0.CO;2-E