---
title: Square Integer Relative Heffter Arrays
url: https://www.emergentmind.com/topics/square-integer-relative-heffter-arrays
type: topic
---

# Square Integer Relative Heffter Arrays

Square integer relative Heffter arrays are \(n\times n\) partially filled arrays over a cyclic group \(\mathbb Z_{2nk+t}\), relative to a subgroup \(J\) of order \(t\), such that each row and each column contains exactly \(k\) filled cells, exactly one of \(x\) or \(-x\) appears for every \(x\in \mathbb Z_{2nk+t}\setminus J\), and every row and column sums to \(0\) in \(\mathbb Z\). In the standard notation they are integer \(H_t(n;k)\), and in the broader \(\lambda\)-fold framework they are precisely integer \({}^1H_t(n;k)\). They generalize ordinary square integer Heffter arrays, recovered at \(t=1\), by replacing the exclusion of \(\{0\}\) with the exclusion of a nontrivial subgroup \(J\) and by replacing the classical half-set of \(\mathbb Z_{2nk+1}\setminus\{0\}\) with a half-set of \(\mathbb Z_{2nk+t}\setminus J\) [2509.09907][2010.10948].

## 1. Definition, support, and the relative condition

For
\[
v=2nk+t,
\]
with \(t\mid 2nk\), let \(J\) be the subgroup of \(\mathbb Z_v\) of order \(t\). A relative Heffter array \(H_t(m,n;h,k)\) over \(\mathbb Z_v\) relative to \(J\) is an \(m\times n\) partially filled array such that each row contains \(h\) filled cells, each column contains \(k\) filled cells, for every \(x\in \mathbb Z_v\setminus J\) either \(x\) or \(-x\) appears, and every row and column sums to \(0\) in \(\mathbb Z_v\). In the square case \(m=n\) and \(h=k\), the notation becomes \(H_t(n;k)\). The prefix \(I\) indicates the integer property: in an \(IH_t(n;k)\), the row and column sums are not merely \(0\pmod v\), but actually \(0\) in \(\mathbb Z\) [2509.09907].

The relative condition is the essential distinction from ordinary Heffter arrays. Ordinary arrays correspond to \(t=1\), so the only excluded element is \(0\). Relative arrays instead omit the subgroup \(J\), and the filled entries form a half-set of \(\mathbb Z_v\setminus J\). In the \(\lambda\)-fold generalization, the same square object becomes \({}^\lambda H_t(n;k)\); if \(\lambda=1\), one recovers the relative Heffter arrays of Costa–Morini–Pasotti–Pellegrini, and if \(\lambda=t=1\), one recovers the classical Heffter arrays [2010.10948].

The support is the set of absolute values of the entries. For square integer relative arrays, a convenient form is
\[
\operatorname{supp}(A)=\left[1,\,nk+\frac t2\right]\setminus \{\ell,2\ell,\dots,\tfrac t2\ell\},
\qquad
\ell=\frac{2nk}{t}+1,
\]
when \(t\) is even; for odd \(t\), the upper bound becomes \(nk+\frac{t-1}{2}\) and the omitted multiples run only to \(\frac{t-1}{2}\ell\). A basic special case is \(t=2\): \(IH_2(m,n;h,k)\) and ordinary \(IH(m,n;h,k)\) have the same support, so \(t=2\) is essentially equivalent to the classical integer Heffter-array case [1910.09921].

## 2. Arithmetic constraints and the even-occupancy square theory

The general square existence problem is controlled first by arithmetic necessities. For ordinary relative integer arrays \(H_t(n;k)\), prior work recalled in the later literature gives the following necessary conditions: if \(t\mid nk\), then
\[
nk\equiv 0\pmod 4
\quad\text{or}\quad
nk\equiv -t\equiv \pm 1\pmod 4;
\]
if \(t=2nk\), then \(k\) must be even; and if \(t\neq 2nk\) and \(t\nmid nk\), then
\[
t+2nk\equiv 0\pmod 8.
\]
These conditions specialize many modular obstructions already visible in the square case [2509.09907].

For even occupancies, the square theory is broad. In the notation \(H_t(n,n;s,s)\), the results for integer relative arrays imply that if
\[
4\le s\le n,\qquad s\equiv 0\pmod 4,
\]
then for every divisor \(t\mid 2ns\) there exists a shiftable integer square relative Heffter array
\[
H_t(n,n;s,s).
\]
If
\[
6\le s\le n,\qquad s\equiv 2\pmod 4,\qquad n\ \text{even},
\]
then for every divisor \(t\mid 2ns\) there again exists a shiftable integer square relative Heffter array \(H_t(n,n;s,s)\). The unresolved square case in that paper is exactly
\[
n\ \text{odd},\qquad s\equiv 2\pmod 4
\]
[1910.09921].

The \(\lambda\)-fold relative theory preserves this square pattern. For integer \({}^\lambda H_t(n,n;s,s)\), existence is proved whenever \(s\equiv 0\pmod 4\), and also whenever \(s\equiv 2\pmod 4\) and \(n\) is even, under the admissibility conditions
\[
\lambda\mid 2ns,\qquad t\mid \frac{2ns}{\lambda}.
\]
In particular, for fully filled square arrays \({}^\lambda H_t(n,n;n,n)\), the paper covers every even \(n\); the remaining unresolved square family is again the odd-order, \(2\bmod 4\) occupancy regime [2010.12333].

## 3. The distinguished family \(H_k(n;k)\)

A central square relative family is obtained by setting the relative parameter equal to the occupancy. In an integer \(H_k(n;k)\), one has
\[
v=2nk+k=k(2n+1),
\]
and the subgroup \(J\le \mathbb Z_v\) of order \(k\) is
\[
J=\{0,2n+1,2(2n+1),\dots,(k-1)(2n+1)\}.
\]
Accordingly, the support is an interval with the multiples of \(2n+1\) removed. This family is the main subject of the first systematic paper on square relative Heffter arrays [1906.03932].

Its existence theory is nearly complete. For \(3\le k\le n\) with \(k\neq 5\), there exists an integer \(H_k(n;k)\) if and only if one of the following holds:
\[
k\ \text{odd and } n\equiv 0,3\pmod 4;
\]
\[
k\equiv 2\pmod 4 \text{ and } n \text{ even};
\]
\[
k\equiv 0\pmod 4.
\]
For \(k=5\), existence is proved when \(n\equiv 3\pmod 4\), nonexistence is proved when \(n\equiv 1,2\pmod 4\), and the case \(n\equiv 0\pmod 4\) is left open. The same paper also proves stronger nonexistence results for some other relative parameters, notably that there is no integer \(H_{3n}(n;3)\) for \(n\ge 3\), and no integer \(H_4(4;3)\) [1906.03932].

The constructions in this family are explicitly diagonal. The paper gives cyclically diagonal base arrays for \(k=3,4,5,6\), and then an extension theorem that enlarges \(k\) by adding suitably designed shiftable arrays on disjoint diagonals. This establishes the central role of square diagonal templates in the relative theory and anticipates later reduction theorems in which square diagonal arrays serve as source objects for more general constructions.

## 4. The \(k=3\) problem, strippability, and the \(t=6\) breakthrough

The most delicate square relative regime presently documented in detail is \(k=3\). A primary transversal in an \(n\times n\) Heffter array is a transversal whose support is \(\{1,\dots,n\}\). An array is shiftable if every row and column contains the same number of positive and negative entries. It is strippable if a primary transversal can be removed leaving a shiftable array. For \(IH_t(n;3)\), the relation is especially tight: if an array has a primary transversal, then it is automatically strippable, because the two non-transversal entries in each row and column must have opposite signs [2509.09907].

This leads to a sharp restriction. If an \(IH_t(n;k)\) with a primary transversal exists, then
\[
\frac{2nk}{n-1}>t.
\]
For \(k=3\), this forces
\[
t\le 6.
\]
Thus strippable square integer relative Heffter arrays with \(k=3\) can occur only for
\[
t\in\{1,2,3,4,5,6\}.
\]
That bound explains the focus on the formerly unresolved \(t=6\) case.

The main recent advance is the construction of two infinite families of strippable
\[
IH_6(n;3)
\]
arrays, one for \(n\equiv 3\pmod 4\) and one for \(n\equiv 0\pmod 4\). Here
\[
v=2nk+t=6n+6,
\]
and the required support is
\[
[1,3n+2]\setminus\{n+1,2n+2\}.
\]
The resulting existence criterion is exact:
\[
IH_6(n;3)\ \text{exists if and only if}\ n\equiv 0,3\pmod 4.
\]
The same paper then combines this theorem with earlier exact results for \(t=1,2,3,n,2n,3n\) to complete the prime-\(n\) existence theory for \(IH_t(n;3)\): for prime \(n\ge 3\),
\[
n\equiv 1\pmod 4
\iff
t\in\{1,2,n,2n\},
\]
and
\[
n\equiv 3\pmod 4
\iff
t\in\{3,6,n,2n\}.
\]
The paper also records that \(t=4,5\) remain incomplete in the strippable setting: there is no strippable \(IH_4(4;3)\), although an \(IH_4(4;3)\) without a primary transversal does exist, and for \(t=5\) the general sufficiency question is still open [2509.09907].

## 5. Construction paradigms

The construction theory of square integer relative Heffter arrays is overwhelmingly explicit. A recurrent language is diagonal placement. For an \(n\times n\) array, the \(i\)-th diagonal is
\[
D_i=\{(i,1),(i+1,2),\dots,(i+n-1,n)\},
\]
with indices mod \(n\), and one often writes
\[
\operatorname{diag}(r,c,s,\Delta_1,\Delta_2,\ell)
\]
to mean
\[
H_{r+i\Delta_1,\;c+i\Delta_1}=s+i\Delta_2
\qquad (i=0,\dots,\ell-1).
\]
This notation is used both in the \(k=3\), \(t=6\) constructions and in the broader \(\lambda\)-fold framework [2509.09907][2010.10948].

For even occupancies, one standard engine is the shiftable \(2\times 2\) block
\[
B_{a,b}=
\begin{array}{|c|c|}
\hline
1 & -(a+1)\\
\hline
-(b+1) & a+b+1\\
\hline
\end{array},
\]
together with its shifts \(B_{a,b}\pm x\). Because each row and column contains the same number of positive and negative entries, these shifts preserve all row and column sums. In the square case \(s\equiv 0\pmod 4\), repeated diagonal placement of such blocks yields shiftable integer relative arrays for every admissible \(t\) [1910.09921].

For the family \(H_k(n;k)\), the key device is a diagonal extension theorem: starting from an integer \(H_k(n;k)\) and a shiftable partially filled square array on disjoint diagonals with the correct support interval, one can shift the auxiliary array and take the union to obtain an integer \(H_{k+h}(n;k+h)\). The base cases \(k=3,4,5,6\) are all constructed explicitly; larger occupancies are then produced by adding \(4\) or \(6\) diagonals at a time [1906.03932].

The \(IH_6(n;3)\) constructions add a further layer of structure. They are cyclically \(3\)-diagonal, with nonempty cells lying exactly on \(D_2\), \(D_1\), and \(D_n\), and they were designed from integer current assignments on ladder graphs, in the spirit of Archdeacon et al. and ultimately Youngs. In that viewpoint, rung currents become the main diagonal entries \(\{1,\dots,n\}\), the lower diagonal is negative, the upper diagonal is positive, and paired off-diagonal entries satisfy
\[
|H_{i,i+1}|+|H_{i+1,i}|=4n+4.
\]
This suggests a close structural relation between square integer relative Heffter arrays and current-graph constructions in topological graph theory [2509.09907].

## 6. Generalizations, applications, and open directions

Square integer relative Heffter arrays occupy a central place inside the \(\lambda\)-fold relative theory. The fundamental transfer theorem states that if there exists a \({}^\alpha H_t(m,n;s,k)\), then for any divisor \(\lambda\) of \(t\) there exists a \({}^{\alpha\lambda}H_{t/\lambda}(m,n;s,k)\). In the square case this reads
\[
H_t(n;k)\Longrightarrow {}^\lambda H_{t/\lambda}(n;k)\qquad (\lambda\mid t).
\]
The same paper emphasizes, however, that integerity need not be preserved under this folding operation. Thus square integer relative arrays are both a special case and a source class for many non-integer \(\lambda\)-fold relative arrays [2010.10948].

When simplicity or global simplicity is available, square relative arrays yield difference families, cyclic decompositions, and surface embeddings. In the \(\lambda\)-fold setting, a simple square \({}^\lambda H_t(n;k)\) produces a
\[
\left(\frac{2nk}{\lambda}+t,\ t,\ C_k,\ \lambda\right)\text{-DF},
\]
and hence cyclic \(k\)-cycle decompositions of
\[
K^\lambda_{\left(\frac{2nk}{\lambda t}+1\right)\times t}.
\]
If row and column orderings are compatible, one obtains a cellular biembedding of the corresponding cycle decompositions into an orientable surface. This topological role is one reason global simplicity has become a recurrent auxiliary objective in the square theory [2010.10948].

A nearby but distinct literature studies square relative non-zero-sum Heffter arrays, written \(NH_t(n;k)\) or \({}^\lambda NH_t(n;k)\). Their support condition is the same relative half-set condition, but the row and column sums are required to be different from \(0\), not equal to \(0\). In the square non-zero-sum setting, complete modular existence theorems are known, and globally simple square examples exist for all \(n\ge k\ge 1\). This is not an integer theory, but it clarifies a frequent source of confusion: zero-sum relative arrays and non-zero-sum relative arrays belong to the same support-combinatorial ecosystem while solving opposite balancing problems [2109.09365].

Several open directions remain. In the square integer \(\lambda\)-fold relative theory with even occupancy, the odd-order case
\[
n\ \text{odd},\qquad s\equiv 2\pmod 4
\]
is not settled [2010.12333]. In the \(k=3\) theory, the strippable cases \(t=4\) and \(t=5\) remain incomplete, even though \(t=6\) is now completely resolved [2509.09907]. More broadly, the very general rectangular case with arbitrary \(\lambda\), arbitrary \(t\), and at least one of the occupancies odd is described as completely open in the \(\lambda\)-fold relative framework [2010.10948]. A plausible implication is that future progress will continue to depend on the same square techniques that already dominate the known theory: diagonal templates, shiftable local blocks, primary transversals, and current-graph or difference-family interpretations.

Source: https://www.emergentmind.com/topics/square-integer-relative-heffter-arrays