### Abstract

For a graph G whose number of edges is divisible by k, let R(G,Z_{k}) denote the minimum integer r such that for every function f: E(K_{r}) ↦ Z_{k} there is a copy G^{1} of G in K_{r} so that Σe∈E(G^{1}) f(e) = 0 (in Z_{k}). We prove that for every integer k_{1} R(K_{n}, Z_{k}) ≤ n + O(k^{3} log k) provided n is sufficiently large as a function of k and k divides ( 2n). If, in addition, k is an odd prime‐power then R(K_{n}, Z_{k}) ≤ n + 2k ‐ 2 and this is tight if k is a prime that divides n. A related result is obtained for hypergraphs. It is further shown that for every graph G on n vertices with an even number of edges R(G,Z_{2}) ≤ n + 2. This estimate is sharp. © 1993 John Wiley & Sons, Inc.

Original language | English (US) |
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Pages (from-to) | 177-192 |

Number of pages | 16 |

Journal | Journal of Graph Theory |

Volume | 17 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1993 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Geometry and Topology

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## Cite this

*Journal of Graph Theory*,

*17*(2), 177-192. https://doi.org/10.1002/jgt.3190170207