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Dedicated to the memory of our friend and colleague Scott Vanstone. AbstractA Heffter array H(m, n; s, t) is an m × n matrix with nonzero entries from Z 2ms+1 such that i) each row contains s filled cells and each column contains t filled cells, ii) every row and column sum to 0, and iii) no element from {x, −x} appears twice. Heffter arrays are useful in embedding the complete graph K 2nm+1 on an orientable surface where the embedding has the property that each edge borders exactly one s−cycle and one t−cycle. Archdeacon, Boothby and Dinitz proved that these arrays can be constructed in the case when s = m, i.e every cell is filled. In this paper we concentrate on square arrays with empty cells where every row sum and every column sum is 0 in Z. We solve most of the instances of this case.

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The Hamilton-Waterloo problem is to determine the existence of a 2-factorization of K 2n+1 in which r of the 2-factors are isomorphic to a given 2-factor R and s of the 2-factors are isomorphic to a given 2-factor S, with r +s = n. In this paper we consider the case when R is a triangle-factor, S is a Hamilton cycle and s = 1. We solve the problem completely except for 14 possible exceptions. This solves a major open case from the 2004 paper of Horak, Nedela, and Rosa.

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 2ms+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.