Square Integer Heffter Arrays with Empty Cells

2014/12/29 by Archdeacon, D. S., Dinitz, J. H., Donovan, D. M. +1 · 3 citations
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1412.8409

Abstract

A 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.

Cited by

• Globally Simple Heffter Arrays H(n;k) with k ≡ 1 \pmod4
• An update on the existence of integer Heffter arrays
• Further constructions of square integer relative Heffter arrays

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