Square integer Heffter arrays with empty cells

Abstract

A Heffter array is an matrix with nonzero entries from such that (i) each row contains filled cells and each column contains filled cells, (ii) every row and column sum to 0, and (iii) no element from appears twice. Heffter arrays are useful in embedding the complete graph 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 , 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 in . We solve most of the instances of this case.