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Square Integer Heffter Arrays with Empty Cells
- Publication Year :
- 2014
-
Abstract
- 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 $K_{2nm+1}$ 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 $\mathbb{Z}$. We solve most of the instances of this case.<br />20 pages, including 2 figures
- Subjects :
- FOS: Mathematics
Mathematics - Combinatorics
Combinatorics (math.CO)
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....4fd826a6c3afbabd4391554a30dabd20