## Abstract

Given a graph

*G*on*n*vertices, for which*m*is it possible to partition the edge set of the*m*-fold complete graph*mK*into copies of_{n}*G*? We show that there is an integer*m*_{0}, which we call the*partition modulus of G*, such that the set*M*(*G*) of values of*m*for which such a partition exists consists of all but finitely many multiples of*m*_{0}. Trivial divisibility conditions derived from*G*give an integer*m*_{1}that divides*m*_{0}; we call the quotient*m*_{0}/*m*_{1}the*partition index of G*. It seems that most graphs*G*have partition index equal to 1, but we give two infinite families of graphs for which this is not true. We also compute*M*(*G*) for various graphs and outline some connections between our problem and the existence of designs of various types.Original language | English |
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Pages (from-to) | 972-983 |

Number of pages | 12 |

Journal | The American Mathematical Monthly |

Volume | 122 |

Issue number | 10 |

DOIs | |

Publication status | Published - Dec 2015 |

## Keywords

- graph, design, affine plane