### Abstract

An integer Heffter array H(m, n; s, t) is an m × n partially filled matrix with entries from the set {±1, ±2, . . ., ±ms} such that i) each row contains s filled cells and each column contains t filled cells, ii) every row and column sums to 0 (in Z), and iii) no two entries agree in absolute value. Heffter arrays are useful for embedding the complete graph K _{2} ms _{+1} on an orientable surface in such a way that each edge lies between a face bounded by an s-cycle and a face bounded by a t-cycle. In 2015, Archdeacon, Dinitz, Donovan and Yazıcı constructed square (i.e. m = n) integer Heffter arrays for many congruence classes. In this paper we construct square integer Heffter arrays for all the cases not found in that paper, completely solving the existence problem for square integer Heffter arrays.

Original language | English |
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Pages (from-to) | 81-93 |

Number of pages | 13 |

Journal | Ars Mathematica Contemporanea |

Volume | 16 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

### Keywords

- Biembedding
- Heffter array

### Cite this

*Ars Mathematica Contemporanea*,

*16*(1), 81-93. https://doi.org/10.26493/1855-3974.1664.4b6