Square Integer Relative Heffter Arrays

• Square integer relative Heffter arrays are n×n partially filled arrays over cyclic groups Z_{2nk+t} with exactly k filled cells per row and column that sum to zero in Z.
• They generalize classical Heffter arrays by excluding a nontrivial subgroup J and using a half-set condition on Z_{2nk+t}\setminus J, maintaining balance in the entries.
• Construction techniques involve diagonal placements, shiftable blocks, and current-graph methods, with open challenges remaining in low occupancy and non-zero-sum frameworks.

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 $\mathbb Z_{2nk+t}$0, and in the broader $\mathbb Z_{2nk+t}$1-fold framework they are precisely integer $\mathbb Z_{2nk+t}$2. They generalize ordinary square integer Heffter arrays, recovered at $\mathbb Z_{2nk+t}$3, by replacing the exclusion of $\mathbb Z_{2nk+t}$4 with the exclusion of a nontrivial subgroup $\mathbb Z_{2nk+t}$5 and by replacing the classical half-set of $\mathbb Z_{2nk+t}$6 with a half-set of $\mathbb Z_{2nk+t}$7 (Donovan et al., 12 Sep 2025, Costa et al., 2020).

1. Definition, support, and the relative condition

For

$\mathbb Z_{2nk+t}$8

with $\mathbb Z_{2nk+t}$9, let $J$0 be the subgroup of $J$1 of order $J$2. A relative Heffter array $J$3 over $J$4 relative to $J$5 is an $J$6 partially filled array such that each row contains $J$7 filled cells, each column contains $J$8 filled cells, for every $J$9 either $t$0 or $t$1 appears, and every row and column sums to $t$2 in $t$3. In the square case $t$4 and $t$5, the notation becomes $t$6. The prefix $t$7 indicates the integer property: in an $t$8, the row and column sums are not merely $t$9, but actually $k$0 in $k$1 (Donovan et al., 12 Sep 2025).

The relative condition is the essential distinction from ordinary Heffter arrays. Ordinary arrays correspond to $k$2, so the only excluded element is $k$3. Relative arrays instead omit the subgroup $k$4, and the filled entries form a half-set of $k$5. In the $k$6-fold generalization, the same square object becomes $k$7; if $k$8, one recovers the relative Heffter arrays of Costa–Morini–Pasotti–Pellegrini, and if $k$9, one recovers the classical Heffter arrays (Costa et al., 2020).

The support is the set of absolute values of the entries. For square integer relative arrays, a convenient form is

$x$0

when $x$1 is even; for odd $x$2, the upper bound becomes $x$3 and the omitted multiples run only to $x$4. A basic special case is $x$5: $x$6 and ordinary $x$7 have the same support, so $x$8 is essentially equivalent to the classical integer Heffter-array case (Morini et al., 2019).

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 $x$9, prior work recalled in the later literature gives the following necessary conditions: if $-x$0, then

$-x$1

if $-x$2, then $-x$3 must be even; and if $-x$4 and $-x$5, then

$-x$6

These conditions specialize many modular obstructions already visible in the square case (Donovan et al., 12 Sep 2025).

For even occupancies, the square theory is broad. In the notation $-x$7, the results for integer relative arrays imply that if

$-x$8

then for every divisor $-x$9 there exists a shiftable integer square relative Heffter array

$x\in \mathbb Z_{2nk+t}\setminus J$0

If

$x\in \mathbb Z_{2nk+t}\setminus J$1

then for every divisor $x\in \mathbb Z_{2nk+t}\setminus J$2 there again exists a shiftable integer square relative Heffter array $x\in \mathbb Z_{2nk+t}\setminus J$3. The unresolved square case in that paper is exactly

$x\in \mathbb Z_{2nk+t}\setminus J$4

(Morini et al., 2019).

The $x\in \mathbb Z_{2nk+t}\setminus J$5-fold relative theory preserves this square pattern. For integer $x\in \mathbb Z_{2nk+t}\setminus J$6, existence is proved whenever $x\in \mathbb Z_{2nk+t}\setminus J$7, and also whenever $x\in \mathbb Z_{2nk+t}\setminus J$8 and $x\in \mathbb Z_{2nk+t}\setminus J$9 is even, under the admissibility conditions

$0$0

In particular, for fully filled square arrays $0$1, the paper covers every even $0$2; the remaining unresolved square family is again the odd-order, $0$3 occupancy regime (Morini et al., 2020).

3. The distinguished family $0$4

A central square relative family is obtained by setting the relative parameter equal to the occupancy. In an integer $0$5, one has

$0$6

and the subgroup $0$7 of order $0$8 is

$0$9

Accordingly, the support is an interval with the multiples of $\mathbb Z$0 removed. This family is the main subject of the first systematic paper on square relative Heffter arrays (Costa et al., 2019).

Its existence theory is nearly complete. For $\mathbb Z$1 with $\mathbb Z$2, there exists an integer $\mathbb Z$3 if and only if one of the following holds: $\mathbb Z$4

$\mathbb Z$5

$\mathbb Z$6

For $\mathbb Z$7, existence is proved when $\mathbb Z$8, nonexistence is proved when $\mathbb Z$9, and the case $\mathbb Z_{2nk+t}$00 is left open. The same paper also proves stronger nonexistence results for some other relative parameters, notably that there is no integer $\mathbb Z_{2nk+t}$01 for $\mathbb Z_{2nk+t}$02, and no integer $\mathbb Z_{2nk+t}$03 (Costa et al., 2019).

The constructions in this family are explicitly diagonal. The paper gives cyclically diagonal base arrays for $\mathbb Z_{2nk+t}$04, and then an extension theorem that enlarges $\mathbb Z_{2nk+t}$05 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 $\mathbb Z_{2nk+t}$06 problem, strippability, and the $\mathbb Z_{2nk+t}$07 breakthrough

The most delicate square relative regime presently documented in detail is $\mathbb Z_{2nk+t}$08. A primary transversal in an $\mathbb Z_{2nk+t}$09 Heffter array is a transversal whose support is $\mathbb Z_{2nk+t}$10. 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 $\mathbb Z_{2nk+t}$11, 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 (Donovan et al., 12 Sep 2025).

This leads to a sharp restriction. If an $\mathbb Z_{2nk+t}$12 with a primary transversal exists, then

$\mathbb Z_{2nk+t}$13

For $\mathbb Z_{2nk+t}$14, this forces

$\mathbb Z_{2nk+t}$15

Thus strippable square integer relative Heffter arrays with $\mathbb Z_{2nk+t}$16 can occur only for

$\mathbb Z_{2nk+t}$17

That bound explains the focus on the formerly unresolved $\mathbb Z_{2nk+t}$18 case.

The main recent advance is the construction of two infinite families of strippable

$\mathbb Z_{2nk+t}$19

arrays, one for $\mathbb Z_{2nk+t}$20 and one for $\mathbb Z_{2nk+t}$21. Here

$\mathbb Z_{2nk+t}$22

and the required support is

$\mathbb Z_{2nk+t}$23

The resulting existence criterion is exact: $\mathbb Z_{2nk+t}$24 The same paper then combines this theorem with earlier exact results for $\mathbb Z_{2nk+t}$25 to complete the prime-$\mathbb Z_{2nk+t}$26 existence theory for $\mathbb Z_{2nk+t}$27: for prime $\mathbb Z_{2nk+t}$28,

$\mathbb Z_{2nk+t}$29

and

$\mathbb Z_{2nk+t}$30

The paper also records that $\mathbb Z_{2nk+t}$31 remain incomplete in the strippable setting: there is no strippable $\mathbb Z_{2nk+t}$32, although an $\mathbb Z_{2nk+t}$33 without a primary transversal does exist, and for $\mathbb Z_{2nk+t}$34 the general sufficiency question is still open (Donovan et al., 12 Sep 2025).

5. Construction paradigms

The construction theory of square integer relative Heffter arrays is overwhelmingly explicit. A recurrent language is diagonal placement. For an $\mathbb Z_{2nk+t}$35 array, the $\mathbb Z_{2nk+t}$36-th diagonal is

$\mathbb Z_{2nk+t}$37

with indices mod $\mathbb Z_{2nk+t}$38, and one often writes

$\mathbb Z_{2nk+t}$39

to mean

$\mathbb Z_{2nk+t}$40

This notation is used both in the $\mathbb Z_{2nk+t}$41, $\mathbb Z_{2nk+t}$42 constructions and in the broader $\mathbb Z_{2nk+t}$43-fold framework (Donovan et al., 12 Sep 2025, Costa et al., 2020).

For even occupancies, one standard engine is the shiftable $\mathbb Z_{2nk+t}$44 block

$\mathbb Z_{2nk+t}$45

together with its shifts $\mathbb Z_{2nk+t}$46. 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 $\mathbb Z_{2nk+t}$47, repeated diagonal placement of such blocks yields shiftable integer relative arrays for every admissible $\mathbb Z_{2nk+t}$48 (Morini et al., 2019).

For the family $\mathbb Z_{2nk+t}$49, the key device is a diagonal extension theorem: starting from an integer $\mathbb Z_{2nk+t}$50 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 $\mathbb Z_{2nk+t}$51. The base cases $\mathbb Z_{2nk+t}$52 are all constructed explicitly; larger occupancies are then produced by adding $\mathbb Z_{2nk+t}$53 or $\mathbb Z_{2nk+t}$54 diagonals at a time (Costa et al., 2019).

The $\mathbb Z_{2nk+t}$55 constructions add a further layer of structure. They are cyclically $\mathbb Z_{2nk+t}$56-diagonal, with nonempty cells lying exactly on $\mathbb Z_{2nk+t}$57, $\mathbb Z_{2nk+t}$58, and $\mathbb Z_{2nk+t}$59, 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 $\mathbb Z_{2nk+t}$60, the lower diagonal is negative, the upper diagonal is positive, and paired off-diagonal entries satisfy

$\mathbb Z_{2nk+t}$61

This suggests a close structural relation between square integer relative Heffter arrays and current-graph constructions in topological graph theory (Donovan et al., 12 Sep 2025).

6. Generalizations, applications, and open directions

Square integer relative Heffter arrays occupy a central place inside the $\mathbb Z_{2nk+t}$62-fold relative theory. The fundamental transfer theorem states that if there exists a $\mathbb Z_{2nk+t}$63, then for any divisor $\mathbb Z_{2nk+t}$64 of $\mathbb Z_{2nk+t}$65 there exists a $\mathbb Z_{2nk+t}$66. In the square case this reads

$\mathbb Z_{2nk+t}$67

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 $\mathbb Z_{2nk+t}$68-fold relative arrays (Costa et al., 2020).

When simplicity or global simplicity is available, square relative arrays yield difference families, cyclic decompositions, and surface embeddings. In the $\mathbb Z_{2nk+t}$69-fold setting, a simple square $\mathbb Z_{2nk+t}$70 produces a

$\mathbb Z_{2nk+t}$71

and hence cyclic $\mathbb Z_{2nk+t}$72-cycle decompositions of

$\mathbb Z_{2nk+t}$73

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 (Costa et al., 2020).

A nearby but distinct literature studies square relative non-zero-sum Heffter arrays, written $\mathbb Z_{2nk+t}$74 or $\mathbb Z_{2nk+t}$75. Their support condition is the same relative half-set condition, but the row and column sums are required to be different from $\mathbb Z_{2nk+t}$76, not equal to $\mathbb Z_{2nk+t}$77. In the square non-zero-sum setting, complete modular existence theorems are known, and globally simple square examples exist for all $\mathbb Z_{2nk+t}$78. 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 (Costa et al., 2021).

Several open directions remain. In the square integer $\mathbb Z_{2nk+t}$79-fold relative theory with even occupancy, the odd-order case

$\mathbb Z_{2nk+t}$80

is not settled (Morini et al., 2020). In the $\mathbb Z_{2nk+t}$81 theory, the strippable cases $\mathbb Z_{2nk+t}$82 and $\mathbb Z_{2nk+t}$83 remain incomplete, even though $\mathbb Z_{2nk+t}$84 is now completely resolved (Donovan et al., 12 Sep 2025). More broadly, the very general rectangular case with arbitrary $\mathbb Z_{2nk+t}$85, arbitrary $\mathbb Z_{2nk+t}$86, and at least one of the occupancies odd is described as completely open in the $\mathbb Z_{2nk+t}$87-fold relative framework (Costa et al., 2020). 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.