Square integer Heffter arrays with empty cells. (English) Zbl 1323.05025
Summary: A Heffter array \(H(m,n;s,t)\) is an \(m \times n\) matrix with nonzero entries from \(\mathbb{Z}_{2ms+1}\) 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 \(K_{2ms+1}\) on an orientable surface where the embedding has the property that each edge borders exactly one \(s\)-cycle and one \(t\)-cycle. D. S. Archdeacon, J. H. Dinitz and T. Boothby [“Tight Heffter arrays exist for all possible orders” (in preparation)] 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 \({\mathbb{Z}}\). We solve most of the instances of this case.
MSC:
| 05B30 | Other designs, configurations |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
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