[Paper Review] Tight Heffter Arrays Exist for all Possible Values: The Research Report

Dan Archdeacon, Tomas Boothby+1 more|arXiv (Cornell University)|Sep 1, 2015

graph theory and CDMA systems5 references3 citations

TL;DR

This paper proves the existence of tight Heffter arrays $H(m,n)$ for all $m,n \geq 3$, establishing that such arrays exist if and only if both dimensions are at least 3. It further shows that integer Heffter arrays exist precisely when $mn \equiv 0,3 \pmod{4}$, and shiftable integer arrays exist exactly when both $m$ and $n$ are even, using block matrix constructions with modular and integer arithmetic to achieve zero row and column sums while avoiding sign-redundant entries.

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.

Motivation & Objective

• To establish the necessary and sufficient conditions for the existence of tight Heffter arrays $H(m,n)$, where every row and column sums to zero modulo $2mn+1$ and no element $x$ or $-x$ appears twice.
• To determine when integer Heffter arrays (with entries in $\{-mn, \dots, mn\}$ summing to zero over $\mathbb{Z}$) exist, showing existence iff $mn \equiv 0,3 \pmod{4}$.
• To characterize shiftable integer Heffter arrays—those with equal numbers of positive and negative entries per row and column—proving they exist iff both $m$ and $n$ are even.
• To provide explicit constructive methods using modular and integer block matrices to generate all such arrays across all valid parameter combinations.

Proposed method

• Constructing $H(m,n)$ arrays via block matrix tiling using modular arithmetic over $\mathbb{Z}_{2mn+1}$, ensuring row and column sums are zero modulo $2mn+1$.
• Using integer block matrices with entries in $\{-mn, \dots, mn\}$ to build integer Heffter arrays, where row and column sums are zero over $\mathbb{Z}$.
• Defining and applying the shift operation $H \pm k$ to transform arrays while preserving row and column sums, enabling support shifting to cover desired integer intervals.
• Employing $9\times 4$, $7\times 4$, and $4\times 4$ tile constructions ($B_r$, $C_r$, $E_0$) to cover disjoint integer intervals and ensure sum-zero properties.
• Combining blocks $A$, $B_r$, $C_r$, and $E$ into a larger matrix structure, with $A$ and $C_r^T$ arranged to maintain sum-zero row and column totals.
• Verifying that the union of interval supports from all blocks covers $[1, mn]$ exactly, ensuring all absolute values from 1 to $mn$ appear once in absolute value.

Experimental results

Research questions

• RQ1 For which values of $m$ and $n$ do tight Heffter arrays $H(m,n)$ exist?
• RQ2 Under what conditions does an integer Heffter array (with integer entries summing to zero over $\mathbb{Z}$) exist?
• RQ3 When can an integer Heffter array be shiftable, i.e., have equal numbers of positive and negative entries per row and column?
• RQ4 Can explicit constructions be provided for all valid $m,n$ satisfying the existence conditions?
• RQ5 How can block matrix methods be used to systematically generate Heffter arrays with zero row and column sums and no sign-redundant entries?

Key findings

• Tight Heffter arrays $H(m,n)$ exist for all $m,n \geq 3$, with no exceptions.
• Integer Heffter arrays exist if and only if $mn \equiv 0,3 \pmod{4}$, which matches the necessary condition from Lemma 1.3.
• Shiftable integer Heffter arrays exist if and only if both $m$ and $n$ are even, as required by the symmetry of positive and negative entries.
• The paper constructs explicit arrays for all valid cases using modular and integer block matrices, covering all integers from 1 to $mn$ exactly once in absolute value.
• The block construction method ensures that all row and column sums are zero in $\mathbb{Z}$, and the support of the array is exactly $\{1,2,\dots,mn\}$.
• The union of interval supports from all blocks $A$, $B_r$, $C_r$, and $E$ covers $[1, mn]$ without overlap, confirming full coverage of required absolute values.