The existence of square integer Heffter arrays

Jeffrey H. Dinitz, Ian M. Wanless

Research output: Contribution to journalArticleResearchpeer-review

11 Citations (Scopus)


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 languageEnglish
Pages (from-to)81-93
Number of pages13
JournalArs Mathematica Contemporanea
Issue number1
Publication statusPublished - 1 Jan 2019


  • Biembedding
  • Heffter array

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