Tight Heffter Arrays Exist for all Possible Values: The Research Report

Abstract

A tight Heffter array H(m,n) is an m x n matrix with nonzero entries from Z_{2mn+1} such that i) the sum of the elements in each row and each column is 0, and ii) no element from {x,-x\ appears twice. We prove that H(m,n) exist if and only if both m and n are at least 3. If all entries are integers of magnitude at most mn satisfying every row and column sum is 0 over the integers and also satisfying ii) we call H an integer Heffter array. We show integer Heffter arrays exist if and only if mn \equiv 0,3 (mod 4). Finally, an integer Heffter array is shiftable if each row and column contains an the same number of positive and negative integers. We show that shiftable integer arrays exists exactly when both $m,n$ are even. This research report contains all of the details of the proofs. It is meant to accompany the journal version of this paper.