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**Covering and packing in graphs. V. Mispacking subcubes in hypercubes.**
*(English)*
Zbl 0645.05058

A node-disjoint packing of a graph G with a subgraph H is a largest collection of disjoint copies of a smaller graph H contained in G; an edge disjoint packing is defined similarly, but no two copies of H have a common edge. Two packing numbers of G with H are defined accordingly. It is easy to determine both of these numbers when H is a subcube of a hypercube G. A mispacking of G with subgraphs H is a minimum maximal collection of disjoint copies of H whose removal from G leaves no subgraph H. Two mispacking numbers of G and H are defined analogously to the packing numbers. Their exact determination is quite difficult but we obtain upper bounds. The covering number of G by a subgraph H is the smallest number of copies of H whose union is all of G. This number is determined for \(G=Q_ n\), \(H=Q_ m\).

### MSC:

05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |

### References:

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[2] | Akiyama, J.; Exoo, G.; Harary, F., Covering and packing in graphs IV: Linear arboricity, Networks, 11, 69-72 (1981) · Zbl 0479.05027 |

[3] | Harary, F., (Graph Theory (1969), Addison-Wesley: Addison-Wesley Reading) · Zbl 0182.57702 |

[4] | Harary, F., Maximum versus minimum invariants for graphs, J. Graph Theory, 7, 275-284 (1983) · Zbl 0515.05053 |

[5] | J. P. Hayes, T. N. Mudge, Q. F. Stout, S. Colley and J. Palmer. Architecture of a hypercube supercomputer. Proc. 1986 Int. Conf. on Parallel Processing; J. P. Hayes, T. N. Mudge, Q. F. Stout, S. Colley and J. Palmer. Architecture of a hypercube supercomputer. Proc. 1986 Int. Conf. on Parallel Processing |

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