Tight Heffter Arrays Exist for all Possible Values

Abstract

A tight Heffter array H(m,n) is an m×n matrix with nonzero entries from Z2mn+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 H has the property that all entries are integers of magnitude at most mn, every row and column sum is 0 over the integers, and H also satisfies (ii), we call H an integer Heffter array. We show integer Heffter arrays exist if and only if mn≡0,3(mod4). Finally, an integer Heffter array is shiftable if each row and column contains the same number of positive and negative integers. We show that shiftable integer arrays exists exactly when both m,n are even.