# A survey of Heffter arrays

Anita Pasotti
DICATAM - Sez. Matematica
Università degli Studi di Brescia
Via Branze 43
I-25123 Brescia Italy
<EMAIL_ADDRESS>
Partially supported by INdAM - GNSAGA Jeffrey H. Dinitz
Dept of Math. and Stat.
University of Vermont
Burlington, VT 05405, USA
<EMAIL_ADDRESS>
###### Abstract

Heffter arrays were introduced by Archdeacon in 2015 as an interesting link
between combinatorial designs and topological graph theory. Since the initial
paper on this topic, there has been a good deal of interest in Heffter arrays
as well as in related topics such as the sequencing of subsets of a group,
biembeddings of cycle systems on a surface, and orthogonal cycle systems. This
survey presents an overview of the current state of the art of this topic. We
begin with an introduction to Heffter arrays for the reader who is unfamiliar
with the subject, then we give a unified and comprehensive presentation of the
major results, showing some proof methods also. This survey also includes
sections on the connections of Heffter arrays to several other combinatorial
objects, such as problems on partial sums and sequenceability, biembedding
graphs on surfaces, difference families and orthogonal graph decompositions.
These sections are followed by a section discussing the variants and
generalizations of Heffter arrays which have been proposed. The survey itself
is complemented by a list of unsolved problems as well as an updated and
complete bibliography.

Dedicated to Doug Stinson on the occasion of his 66th birthday.

##  1 Introduction and Background

The notion of a Heffter array was introduced by Archdeacon [7] in 2015. The
original motivation was to aid in an embedding problem in topological graph
theory, but they were also introduced as a very interesting combinatorial
design object. The name is derived from their connection to Heffter’s first
difference problem, a problem related to the construction of Steiner triple
systems [24]. Specifically, in 1896 Heffter [49] introduced his famous _First
Difference Problem_ which asks if, given $v\equiv 1\pmod{6}$, the set
$\left\\{1,2,\ldots,\frac{v-1}{2}\right\\}$ can be partitioned into
$\frac{v-1}{6}$ triples $\\{x,y,z\\}$ such that either $x+y=z$ or $x+y+z=v$.
The problem was eventually solved in 1939 by Peltesohn [68]. We will show
later in this survey that this problem is essentially solved (with an even
stronger property) by a Heffter array with 3 rows and $\frac{v-1}{6}$ columns.

We begin with some definitions. Let $\mathbb{Z}_{v}$ be the cyclic group of
odd order $v$ whose elements are denoted by $0$ and $\pm i$ where
$i=1,\ldots,\frac{v-1}{2}$. A _half-set_ $L$ of $\mathbb{Z}_{v}$ is a subset
of $\mathbb{Z}_{v}\setminus\\{0\\}$ of size $\frac{v-1}{2}$ that contains
exactly one of each pair $\\{x,-x\\}$. A _Heffter system_ $D(v,k)$, on a half-
set $L$ of $\mathbb{Z}_{v}$, is a partition of $L$ into parts of size $k$ such
that the elements in each part sum to $0$ modulo $v$, see [7, 60]. Clearly a
necessary condition for the existence of such a system is $v\equiv
1\pmod{2k}$.

###### Example 1.1

A Heffter system $D(31,5)$:

$\\{\\{6,7,-10,-4,1\\},\\{-9,5,2,-11,13\\},\\{3,-12,8,15,-14\\}\\}.$

It is easy to see that starting from a Heffter system $D(v,3)$ one immediately
gets a solution to the Heffter’s First Difference Problem, as shown in the
following example.

###### Example 1.2

A Heffter system $D(31,3)$:

$\\{\\{6,3,-9\\},\\{7,5,-12\\},\\{2,8,-10\\},\\{-4,-11,15\\},\\{1,13,-14\\}\\}.$

Then $\\{\\{6,3,9\\},\\{7,5,12\\},\\{2,8,10\\},\\{4,11,15\\},\\{1,13,14\\}\\}$
is a solution to the Heffter’s First Difference Problem with $v=31$. Using
this Heffter system we get the following base blocks
$\\{\\{0,6,9\\},\\{0,7,12\\},\\{0,2,10\\},\\{0,4,15\\},\\{0,1,14\\}\\}$ which
can then be developed in $\mathbb{Z}_{31}$ to construct a Steiner triple
system of order 31. A detailed discussion of the construction of cycle systems
from Heffter systems can be found in Section 5.

Two Heffter systems $D_{m}=D(2mn+1,m)$ and $D_{n}=D(2mn+1,n)$ on the same
half-set $L$ of $\mathbb{Z}_{2mn+1}$ are _orthogonal_ if each part of $D_{m}$
intersects each part of $D_{n}$ in exactly one element, see [7]. By a direct
check one can see that the Heffter systems of Examples 1.1 and 1.2 are
orthogonal.

In [7] Archdeacon defined a _Heffter array_ $H(m,n)$ as an $m\times n$ array
whose rows form a $D(2mn+1,n)$ and whose columns form a $D(2mn+1,m)$; these
are called the row and column Heffter systems, respectively. Note that a
Heffter array $H(m,n)$ is equivalent to a pair of orthogonal Heffter systems.
An $H(m,n)$ gives two orthogonal Heffter systems by definition; its associated
row system and its associated column system. Conversely, two orthogonal
Heffter systems $D(2mn+1,m)$ and $D(2mn+1,n)$ give rise to an $H(m,n)$ whose
cell of indices $i,j$ contains the only element which the $i$-th part of the
first Heffter system shares with the $j$-th part of the second one.

###### Example 1.3

Starting from the orthogonal Heffter systems of Examples 1.1 and 1.2 we obtain
the following Heffter array $H(3,5)$ whose elements belong to
$\mathbb{Z}_{31}$:

$\begin{array}[]{|r|r|r|r|r|}\hline\cr 6&7&-10&-4&1\\\
\hline\cr-9&5&2&-11&13\\\ \hline\cr 3&-12&8&15&-14\\\ \hline\cr\end{array}$

In the same paper Archdeacon proposed a variation of Heffter arrays which
allows for some empty cells. Two Heffter systems $D_{h}=D(2mh+1,h)$ and
$D_{k}=D(2nk+1,k)$ on the same half-set of size $mh=nk$ are _sub-orthogonal_
if each part of $D_{h}$ intersects each part of $D_{k}$ in at most one element
(see [5] for information on sub-orthogonal factorizations). We note here that
the term sub-orthogonal has not been widely adopted, hence for the remainder
of this survey we will use orthogonal to denote either orthogonal or sub-
orthogonal. As before, we can form an $m\times n$ array, denoted by
$H(m,n;h,k)$, whose cell of indices $i,j$ either contains the common element
in the $i$-th part of $D_{h}$ and the $j$-th part of $D_{k}$, if any, or is
empty otherwise. The $H(m,n)$ defined above is nothing but a $H(m,n;n,m)$.

In other words, when empty cells are allowed, a Heffter array can be defined
as follows (this is now the standard definition):

###### Definition 1.4

A _Heffter array_ $H(m,n;h,k)$ is an $m\times n$ matrix with entries from
$\mathbb{Z}_{2nk+1}$ such that:

  * (a)

each row contains $h$ filled cells and each column contains $k$ filled cells;

  * (b)

for every $x\in\mathbb{Z}_{2nk+1}\setminus\\{0\\}$, either $x$ or $-x$ appears
in the array;

  * (c)

the elements in every row and column sum to $0$ in $\mathbb{Z}_{2nk+1}$.

###### Example 1.5

An $H(6,12;8,4)$ over $\mathbb{Z}_{97}$:

$\begin{array}[]{|r|r|r|r|r|r|r|r|r|r|r|r|}\hline\cr&&-1&2&5&-6&&&-25&26&29&-30\\\
\hline\cr&&3&-4&-7&8&&&27&-28&-31&32\\\ \hline\cr
9&-10&&&-13&14&33&-34&&&-37&38\\\ \hline\cr-11&12&&&15&-16&-35&36&&&39&-40\\\
\hline\cr-17&18&21&-22&&&-41&42&45&-46&&\\\ \hline\cr
19&-20&-23&24&&&43&-44&-47&48&&\\\ \hline\cr\end{array}$

Note that by the pigeon-hole principle zero cannot appear in the array, hence
Condition (b) of Definition 1.4 is equivalent to requiring that the elements
appearing in the array form a half-set of $\mathbb{Z}_{2nk+1}$. It is easy to
see that trivial necessary conditions for the existence of an $H(m,n;h,k)$ are
$mh=nk$, $3\leq k\leq m$, and $3\leq h\leq n$.

If a Heffter array $H(m,n;h,k)$ is square, that is $m=n$, then necessarily
$h=k$. In this case it is a $H(n,n;k,k)$ and is denoted by $H(n;k)$. And just
to clarify, a $H(n,k)$ (note the comma in place of the semicolon) is a Heffter
array with no empty cells and with $n$ rows and $k$ columns, or a
$H(n,k;k,n)$.

###### Example 1.6

An $H(5;4)$ over $\mathbb{Z}_{41}$:

$\begin{array}[]{|r|r|r|r|r|}\hline\cr&17&-8&-14&5\\\ \hline\cr
1&&18&-9&-10\\\ \hline\cr-6&2&&19&-15\\\ \hline\cr-11&-12&3&&20\\\ \hline\cr
16&-7&-13&4&\\\ \hline\cr\end{array}$

###### Definition 1.7

A Heffter array is an _integer Heffter array_ if condition (c) of Definition
1.4 is strengthened so that the elements in every row and every column, seen
as integers in $\pm\\{1,2,\ldots,nk\\}$, sum to zero in $\mathbb{Z}$.

Note that the arrays of Examples 1.3, 1.5 and 1.6 are integer Heffter arrays.
In the following example we present a non-integer Heffter array.

###### Example 1.8

A non-integer $H(7;3)$ over $\mathbb{Z}_{43}$:

$\begin{array}[]{|r|r|r|r|r|r|r|}\hline\cr 15&-13&-2&&&&\\\
\hline\cr-11&14&&-3&&&\\\ \hline\cr-4&&-8&12&&&\\\ \hline\cr&-1&10&-9&&&\\\
\hline\cr&&&&5&21&17\\\ \hline\cr&&&&18&6&19\\\ \hline\cr&&&&20&16&7\\\
\hline\cr\end{array}$

A class of Heffter arrays of fundamental importance and which are particularly
useful in recursive constructions of are so called _shiftable_ arrays, see
[9]. An $n\times n$ array $A$ (possibly with empty cells) whose elements are
integers is shiftable if each row and each column of $A$ contains the same
number of positive and negative numbers. Given a shiftable array $A$ and a
nonnegative integer $x$, let $A\pm x$ denote the array where $x$ is added to
all the positive entries in $A$ and $-x$ is added to all the negative entries.
Define the support of $A$ as the set containing the absolute value of the
elements appearing in $A$. Note that if $A$ is shiftable with support $S$ and
$x$ is a nonnegative integer, then $A\pm x$ has the same row and column sums
as $A$ and has support $S+x$. In the case of a shiftable integer Heffter array
$H(m,n;h,k)$, the array $H(m,n;h,k)\pm x$ has row and column sums equal to
zero and its support is $\\{1+x,2+x,\ldots,nk+x\\}$. The arrays of Examples
1.5 and 1.6 are shiftable. The existence of shiftable square Heffter arrays
was established by constructive proofs in [9].

###### Theorem 1.9

[9] There exists a shiftable integer $H(n;k)$ if and only if $k$ is even and
$nk\equiv 0\pmod{4}$.

Many of the known square Heffter arrays have a diagonal structure which has
been shown to be useful for recursive constructions and for the applications
to biembeddings. In addition, this diagonal structure has been extremely
useful in computer searches for Heffter arrays. Given a square array of order
$n$, for $i=1,\ldots,n$, the $i$-th diagonal is defined by
$D_{i}=\\{(i,1),(i+1,2),\ldots,(i-1,n)\\}$. All the arithmetic on row and
column indices is performed modulo $n$, where the set of reduced residues is
$\\{1,2,\ldots,n\\}$. The diagonals $D_{i},D_{i+1},\ldots,D_{i+k-1}$ are $k$
consecutive diagonals. Given $n\geq k\geq 1$, a partially filled array $A$ of
order $n$ is $k$-_diagonal_ if its nonempty cells are exactly those of $k$
diagonals. Moreover, if these diagonals are consecutive, $A$ is said to be
_cyclically_ $k$-_diagonal_. The array of Example 1.6 is cyclically
$4$-diagonal.

Shiftable cyclically $4$-diagonal $H(n;4)$ are constructed in [9] for all
$n\geq 4$. In view of their particular structure, these arrays can be used to
add four filled cells per row and column to an existing Heffter array $H$ if
$H$ contains four consecutive diagonals of empty cells. This is described in
the following lemma and example.

###### Lemma 1.10

[9] If there exists an integer Heffter array $H(n;k)$ which has $s$ disjoint
sets of four consecutive empty diagonals, then there exists an integer
$H(n;k+4s)$. Furthermore, if the $H(n;k)$ is shiftable, then the $H(n;k+4s)$
is shiftable too.

###### Example 1.11

The following are a cyclically $3$-diagonal $H(8;3)$, say $H$, and a shiftable
$H(8;4)$, say $A$:

$H=\begin{array}[]{|r|r|r|r|r|r|r|r|}\hline\cr 8&16&&&&&&-24\\\
\hline\cr-17&-6&23&&&&&\\\ \hline\cr&-10&-5&15&&&&\\\
\hline\cr&&-18&7&11&&&\\\ \hline\cr&&&-22&3&19&&\\\ \hline\cr&&&&-14&2&12&\\\
\hline\cr&&&&&-21&1&20\\\ \hline\cr 9&&&&&&-13&4\\\ \hline\cr\end{array}$

A= $\begin{array}[]{|r|r|r|r|r|r|r|r|}\hline\cr&&&1&-3&-5&7&\\\
\hline\cr&&&-4&2&8&-6&\\\ \hline\cr 15&&&&&9&-11&-13\\\
\hline\cr-14&&&&&-12&10&16\\\ \hline\cr-19&-21&23&&&&&17\\\ \hline\cr
18&24&-22&&&&&-20\\\ \hline\cr&25&-27&-29&31&&&\\\
\hline\cr&-28&26&32&-30&&&\\\ \hline\cr\end{array}$

Using $H$ and $A\pm 24$ one can construct the following $H(8;7)$ over
$\mathbb{Z}_{113}$:

$\begin{array}[]{|r|r|r|r|r|r|r|r|}\hline\cr 8&16&&25&-27&-29&31&-24\\\
\hline\cr-17&-6&23&-28&26&32&-30&\\\ \hline\cr 39&-10&-5&15&&33&-35&-37\\\
\hline\cr-38&&-18&7&11&-36&34&40\\\ \hline\cr-43&-45&47&-22&3&19&&41\\\
\hline\cr 42&48&-46&&-14&2&12&-44\\\ \hline\cr&49&-51&-53&55&-21&1&20\\\
\hline\cr 9&-52&50&56&-54&&-13&4\\\ \hline\cr\end{array}$

Concerning the search for Heffter arrays constructed from diagonals, a useful
notation was introduced in [40] and has been used in several subsequent papers
including [31, 34, 35, 37]. All the constructions in [40] were based on
filling in the cells of a fixed collection of diagonals. To aid in the
constructions the authors defined the following procedure for filling a
sequence of cells on a diagonal. It is termed diag and it has six parameters,
as follows.

In an $n\times n$ array $A$ the procedure
$diag(r,c,s,\Delta_{1},\Delta_{2},\ell)$ installs the entries of an array $A$
as follows:

$A[r+i\Delta_{1},c+i\Delta_{1}]=s+i\Delta_{2}\mbox{ \ for \
}i=0,1,\dots,\ell-1.$

Here all arithmetic on the row and column indices is performed modulo $n$,
where the set of reduced residues is $\\{1,2,\ldots,n\\}$. The following
summarizes the parameters used in the $diag$ procedure:

$\bullet$ |  $r$ denotes the starting row,
---|---
$\bullet$ |  $c$ denotes the starting column,
$\bullet$ |  $s$ denotes the starting symbol,
$\bullet$ |  $\Delta_{1}$ denotes how much the row and column indices change at each step,
$\bullet$ |  $\Delta_{2}$ denotes how much the symbol changes at each step, and
$\bullet$ |  $\ell$ is the length of the chain.

The following example shows the use of the above definition.

###### Example 1.12

[40] The array below is a shiftable integer Heffter array $H(11;4)$ where the
filled cells are contained in two sets of consecutive diagonals and it is
constructed via the following procedures:

$\begin{array}[]{ll}diag(4,1,1,1,2,11);&diag(5,1,-2,1,-2,11);\\\
diag(4,7,-23,1,-2,11);&diag(5,7,24,1,2,11).\end{array}$

$\begin{array}[]{|r|r|r|r|r|r|r|r|r|r|r|}\hline\cr&&38&-39&&&&-16&17&&\\\
\hline\cr&&&40&-41&&&&-18&19&\\\ \hline\cr&&&&42&-43&&&&-20&21\\\ \hline\cr
1&&&&&44&-23&&&&-22\\\ \hline\cr-2&3&&&&&24&-25&&&\\\
\hline\cr&-4&5&&&&&26&-27&&\\\ \hline\cr&&-6&7&&&&&28&-29&\\\
\hline\cr&&&-8&9&&&&&30&-31\\\ \hline\cr-33&&&&-10&11&&&&&32\\\ \hline\cr
34&-35&&&&-12&13&&&&\\\ \hline\cr&36&-37&&&&-14&15&&&\\\ \hline\cr\end{array}$

Concerning the existence problem, the first classes of Heffter arrays which
were systematically studied were (1) square arrays and (2) Heffter arrays with
no empty cells (termed tight Heffter arrays). Square integer Heffter arrays
were considered in [9]. In that paper the authors proved the following almost
complete result. In their proofs Lemma 1.10 plays a fundamental role.

###### Theorem 1.13

[9] If an integer $H(n;k)$ exists, then necessarily $3\leq k\leq n$ and
$nk\equiv 0,3\pmod{4}$. Furthermore, these conditions are sufficient except
possibly when $n\equiv 0\ \mbox{or}\ 3\pmod{4}$ and $k\equiv 1\pmod{4}$.

It should be noted that [9] also contains partial results when $n\equiv 0\
\mbox{or}\ 3\pmod{4}$ and $k\equiv 1$ (mod $4$). These missing cases were
entirely solved in [40] using the $diag$ method described above. Thus we have
a complete constructive solution for the integer square case. This result is
summarized in the following theorem.

###### Theorem 1.14

[9, 40] There exists an integer $H(n;k)$ if and only if $3\leq k\leq n$ and
$nk\equiv 0,3\pmod{4}$.

For all congruences classes of $n$ and $k$ for which an integer Heffter array
necessarily cannot exist a non-integer Heffter array $H(n;k)$ was proven to
exist in [22]. Hence we can state the following.

###### Theorem 1.15

[9, 22, 40] There exists an $H(n;k)$ if and only if $3\leq k\leq n$.

Table 1 details the papers where existence is proven for each congruence class
of $n$ and $k$ modulo 4.

$n\backslash k$ | 0 | 1 | 2 | 3
---|---|---|---|---
0 | [9] | [9, 40] | [9] | [9]
1 | [9] | [22] | [22] | [9]
2 | [9] | [22] | [9] | [22]
3 | [9] | [9, 40] | [22] | [22]
Table 1: Articles which prove the existence of square Heffter arrays $H(n;k)$.

We now turn to the case of rectangular Heffter arrays without empty cells
$H(m,n)$, or so-called tight Heffter arrays. A complete constructive solution
is presented in [8]. Historically speaking, it was this paper that introduced
the notion of shiftable Heffter arrays. The main theorem of that paper is
given next.

###### Theorem 1.16

[8] There exists an $H(m,n)$ if and only if $m,n\geq 3$. Moreover, these
arrays are integer if and only if $mn\equiv 0,3\pmod{4}$.

It has not been proven that integer rectangular arrays with empty cells exist
for all possible orders. We note here that the necesary conditions for the
existence of an integer $H(m,n;h,k)$ are that $3\leq h\leq n$, $3\leq k\leq
m$, $mh=nk$ and $nk\equiv 0,3\pmod{4}$. Recent partial results for integer
rectangular arrays with empty cells can be found in [65]. We summarize these
results in the following two theorems.

###### Theorem 1.17

[65] Let $m,n,h,k$ be such that $3\leq h\leq n$, $3\leq k\leq m$ and $mh=nk$.
Set $d=\gcd(h,k)$. There exists an integer $H(m,n;h,k)$ in each of the
following cases:

  * $(\rm{1})$

$d\equiv 0\pmod{4}$;

  * $(\rm{2})$

$d\equiv 1\pmod{4}$ with $d\geq 5$ and $nk\equiv 3\pmod{4}$;

  * $(\rm{3})$

$d\equiv 2\pmod{4}$ and $nk\equiv 0\pmod{4}$;

  * $(\rm{4})$

$d\equiv 3\pmod{4}$ and $nk\equiv 0,3\pmod{4}$.

###### Theorem 1.18

[65] Let $m,n,h,k$ be such that $3\leq h\leq n$, $3\leq k\leq m$ and $mh=nk$.
If $h\equiv 0\pmod{4}$ and $k\neq 5$ is odd, then there exists an integer
$H(m,n;h,k)$.

This survey is organized as follows. In Section 2 we discuss results on simple
and globally simple Heffter arrays. Then in Section 3 we give the application
of Heffter arrays to biembedding complete graphs on topological surfaces. In
Section 4 we consider the problem of sequencing subsets of a group. Section 5
gives results concerning the connection between Heffter arrays, difference
families and graph decompositions and in Section 6 we survey results
concerning variants and generalizations of Heffter arrays. Finally in Section
7 we give some open problems related to Heffter arrays and the other objects
covered in this survey.

##  2 Simple Heffter Arrays

In [7], Archdeacon showed that Heffter arrays can be used to construct
(simple) cycle decompositions of the complete graph if they satisfy an
additional condition, called _simplicity_. This concept is also related to
several open problems on partial sums and sequenceability, these questions
will be discussed in the Section 4.

Let $T$ be a finite subset of an additive group $G$. Given an ordering
$\omega=(t_{1},t_{2},\ldots,t_{k})$ of the elements in $T$, let
$s_{i}=\sum_{j=1}^{i}t_{j}$ be the $i$-th partial sum of $\omega$. The
ordering $\omega$ is said to be _simple_ if its partial sums
$s_{1},\ldots,s_{k}$ are pairwise distinct. We point out that, if $\sum_{t\in
T}t=0$, then an ordering $\omega$ of the elements in $T$ is simple if and only
if it does not contain any proper subsequence that sums to $0$. We also note
that if $\omega=(t_{1},t_{2},\ldots,t_{k})$ is a simple ordering so is
$\omega^{-1}=(t_{k},t_{k-1},\ldots,t_{1})$.

In the following, with a little abuse of notation, we will identify each row
and column of an $H(m,n;h,k)$ with the subset of $\mathbb{Z}_{2nk+1}$ of size
$h$ and $k$, respectively, whose elements are the entries of the nonempty
cells of such a row or column. For example we can view the first row of the
$H(8;7)$ of Example 1.11 as the subset $R_{1}=\\{8,16,25,-27,-29,31,-24\\}$ of
$\mathbb{Z}_{113}$.

###### Definition 2.1

An $H(m,n;h,k)$ is _simple_ if each row and each column admits a simple
ordering.

It is easy to see that since each row and each column of an $H(m,n;h,k)$ does
not contain $0$ or $2$-subsets of the form $\\{x,-x\\}$, and is such that
$s_{h}=0$ and $s_{k}=0$, respectively, then every $H(m,n;h,k)$ with $h,k\leq
5$ is simple (indeed any subset of size $k\leq 10$ with these properties is
simple [3]).

###### Example 2.2

The $H(8;7)$ of Example 1.11 is simple. To verify this property we need to
provide a simple ordering for each row and each column. One can check that the
$\omega_{i}$’s are simple orderings of the rows and the $\nu_{i}$’s are simple
orderings of the columns:

$\begin{array}[]{lcl}\omega_{1}=(8,25,16,-27,-29,31,-24);&&\nu_{1}=(8,39,-17,-38,-43,42,9);\\\
\omega_{2}=(-17,-6,-28,23,26,32,-30);&&\nu_{2}=(16,-6,-45,-10,48,49,-52);\\\
\omega_{3}=(39,-10,-5,33,15,-35,-37);&&\nu_{3}=(23,-5,47,-18,-46,-51,50);\\\
\omega_{4}=(-38,-18,7,-36,11,34,40);&&\nu_{4}=(25,-28,15,7,-53,-22,56);\\\
\omega_{5}=(-43,-45,47,-22,3,41,19);&&\nu_{5}=(-27,26,11,3,55,-14,-54);\\\
\omega_{6}=(42,48,-46,-14,2,-44,12);&&\nu_{6}=(-21,32,33,-36,19,2,-29);\\\
\omega_{7}=(20,-51,-53,55,-21,1,49);&&\nu_{7}=(-13,-30,-35,34,12,1,31);\\\
\omega_{8}=(-52,9,50,56,-54,-13,4);&&\nu_{8}=(-37,-24,40,41,-44,20,4).\end{array}$

By _natural ordering_ of a row (column, respectively) one means the ordering
from left to right (from top to bottom, respectively). Since the natural
ordering of each row and each column of the $H(8;7)$ of Example 1.11 contains
a proper subsequence that sums to zero (this immediately follows by its
construction), the natural ordering cannot be simple for any row and any
column. Clearly, for larger $n$ and $k$ it is more difficult and tedious to
provide explicit simple orderings for rows and columns of an $H(m,n;h,k)$. For
this reason, in [33] the authors considered Heffter arrays which are simple
with respect to the natural ordering of rows and columns and proposed the
following definition.

###### Definition 2.3

[33] A Heffter array $H(m,n;h,k)$ is _globally simple_ if the natural ordering
of each row and each column is simple.

A globally simple $H(m,n;h,k)$ is denoted by $SH(m,n;h,k)$ and a square
globally simple $H(n;k)$ is denoted by $SH(n;k)$.

By above arguments, we have that the $H(8;7)$ of Example 1.11 is simple but
not globally simple. However a globally simple $H(8;7)$ does indeed exist and
is presented in the following example.

###### Example 2.4

A globally simple $H(8;7)$:

$4$ | $35$ | $-45$ | $46$ |  | $20$ | $-36$ | $-24$
---|---|---|---|---|---|---|---
$48$ | $-5$ | $23$ | $-47$ | $-18$ |  | $37$ | $-38$
$-32$ | $-10$ | $-6$ | $31$ | $-41$ | $42$ |  | $16$
$33$ | $-34$ | $44$ | $3$ | $11$ | $-43$ | $-14$ |
| $15$ | $-28$ | $-22$ | $7$ | $27$ | $-53$ | $54$
$-13$ |  | $29$ | $-30$ | $56$ | $1$ | $12$ | $-55$
$-49$ | $50$ |  | $19$ | $-40$ | $-21$ | $2$ | $39$
$9$ | $-51$ | $-17$ |  | $25$ | $-26$ | $52$ | $8$

The Heffter arrays constructed in [9, 40], in general, are not globally simple
and easy modifications of the existing constructions seem not to produce
globally simple arrays. One exception to this are the $SH(n;6)$’s constructed
in [33, Proposition 4.1] which were obtained by switching the first two
columns of the matrices given in [9, Theorem 2.1]. Globally simple Heffter
arrays are more elegant than simple ones but also more difficult to construct,
in fact there are few papers on these arrays, see [21, 23, 33, 39]. The
following two theorems summarize the results on square globally simple integer
Heffter arrays. The first deals with the case when $k\leq 10$ and the second
considers the existence of several of the congruence classes.

###### Theorem 2.5

[33] Let $3\leq k\leq 10$. There exists an integer globally simple $H(n;k)$ if
and only if $n\geq k$ and $nk\equiv 0,3\pmod{4}$.

###### Theorem 2.6

[21] Let $n\geq k\geq 3$. There exists an integer globally simple $H(n;k)$ in
each of the following cases:

  * $(\rm{1})$

$k\equiv 0\pmod{4}$;

  * $(\rm{2})$

$n\equiv 1\pmod{4}$ and $k\equiv 3\pmod{4}$;

  * $(\rm{3})$

$n\equiv 0\pmod{4}$, $k\equiv 3\pmod{4}$ and $n>>k$.

In the rectangular case we have the following theorem for Heffter arrays with
either three or five rows.

###### Theorem 2.7

[39] There exists a globally simple $H(3,n)$ for every $n\geq 3$ and a
globally simple $H(5,n)$ for all $3\leq n\leq 100$.

##  3 Applications to biembeddings

One of the main reasons that Archdeacon introduced the notion of Heffter
arrays is their connection with biembedding cycle systems on a topological
surface. Several papers focus on this application, see, for instance [23, 33,
36, 39]. We emphasize that there has been previous work done in the area of
biembedding cycle systems, especially triple systems, see [41, 42, 45, 46]. It
is interesting to note that some of the earliest examples of Heffter arrays
are derived from direct constructions of biembeddings. In particular, in [9]
the authors used current graphs (developed by Youngs [72]) based on Möbius
ladders with $n=4m+1$ rungs and on cylindrical ladders with $n=4m$ rungs to
construct diagonal Heffter arrays $H(n;3)$. So not only do Heffter arrays
yield biembeddings (as will be detailed in this section), but biembeddings
have been used to make Heffter arrays.

We recall some basic definitions, see [61] and also note that [48] and [62]
are excellent references for graphs on surfaces. An _embedding_ of a graph (or
multigraph) $\Gamma$ in a surface $\Sigma$ is a continuous injective mapping
$\psi:\Gamma\to\Sigma$, where $\Gamma$ is viewed with the usual topology as
$1$-dimensional simplicial complex. The connected components of
$\Sigma\setminus\psi(\Gamma)$ are called $\psi$-_faces_. If each $\psi$-face
is homeomorphic to an open disc, then the embedding $\psi$ is said to be
_cellular_. In this context, we say that two embeddings
$\psi:\Gamma\rightarrow\Sigma$ and
$\psi^{\prime}:\Gamma^{\prime}\rightarrow\Sigma^{\prime}$ are isomorphic if
and only if there is a graph isomorphism
$\sigma:\Gamma\rightarrow\Gamma^{\prime}$ such that $\sigma(F)$ is a
$\psi^{\prime}$-face if and only if $F$ is a $\psi$-face. A circuit
decomposition of a graph $\Gamma$ is a collection of edge-disjoint subgraphs
of $\Gamma$ such that every edge of $\Gamma$ belongs to exactly one circuit.

###### Definition 3.1

A _biembedding_ of two circuit decompositions $\mathcal{D}$ and
$\mathcal{D}^{\prime}$ of a simple graph $\Gamma$ is a face $2$-colorable
embedding of $\Gamma$ in which one color class is comprised of the circuits in
$\mathcal{D}$ and the other class contains exactly the circuits in
$\mathcal{D}^{\prime}$.

If the faces are cycles the biembedding is said to be _simple_. When one is
not interested in the circuits, we simply speak of a biembedding of the graph
$\Gamma$. Equivalently, a biembedding of a simple graph $\Gamma$ is a face
$2$-colorable embedding of $\Gamma$.

###### Example 3.2

A biembedding of the complete graph $K_{7}$ on the torus. Note that the black
cells are a Steiner triple system (a circuit decomposition of $K_{7}$ into
triples) and the white cells are also a Steiner triple system.

In order to state our main theorem connecting Heffter arrays to biembeddings
(Theorem 3.11), we need the notion of compatible orderings of the rows and
columns of the Heffter array.

###### Definition 3.3

Let $A$ be a partially filled array with $F$ filled cells whose elements are
pairwise distinct. Let $\omega_{r}$ ($\omega_{c}$, respectively) be any
ordering of the rows (columns, respectively) of $A$. Then $\omega_{r}$ and
$\omega_{c}$ are _compatible_ orderings if $\omega_{r}\circ\omega_{c}$ is a
cycle of order $F$.

###### Example 3.4

Given the $H(3,5)$ from Example 1.3, use the natural ordering of the rows and
columns to get row cycles
$\omega_{r}=(6,7,{-10},{-4},1)(-9,5,2,{-11},13)(3,{-12},8,15,{-14})$ and the
column cycles
$\omega_{c}=(6,{-9},3)(7,5,{-12})({-10},2,8)({-4},{-11},15)(1,13,{-14})$. Then
we see that
$\omega_{r}\circ\omega_{c}=(6,5,8,{-4},13,3,7,2,15,1,{-9},{-12},{-10},{-11},{-14})$
is a single cycle (of order 15) and hence $\omega_{r}$ and $\omega_{c}$ are
compatible orderings.

We will discuss the connection between compatible orderings and biembeddings
below. It is because of this connection that the existence of compatible
orderings for a given array has been considered in several papers, see [23,
26, 33, 39]. The following theorem details the necessary conditions for there
to exist orderings $\omega_{r}$ and $\omega_{c}$ of the row and column Heffter
systems of a Heffter array that are compatible.

###### Theorem 3.5

[23, 39] If there exist compatible orderings $\omega_{r}$ and $\omega_{c}$ for
a Heffter array $H(m,n;h,k)$, then either:

  * (1)

$m,n,h$ and $k$ are all odd;

  * (2)

$m$ is odd, $n$ is even and $h$ is even; or

  * (3)

$m$ is even, $n$ is odd and $k$ is even.

Hence, if a square Heffter array $H(n;k)$ admits compatible orderings, then
$nk$ is odd. This condition is also sufficient when $k$ is small and the array
is cyclically $k$-diagonal. In fact the following holds:

###### Proposition 3.6

[26] Let $3\leq k<200$ be an integer and let $A$ be a cyclically $k$-diagonal
Heffter array of size $n\geq k$. Then there exist two compatible orderings
$\omega_{r}$ and $\omega_{c}$ of the rows and the columns of $A$ if and only
if $n$ and $k$ are both odd.

In view of Theorem 3.5, in order to obtain results on biembeddings, several
authors have focused on square Heffter arrays with $n$ and $k$ odd. The
following results pertain to the existence of compatible orderings in the case
of cyclically $k$-diagonal Heffter arrays $H(n;k)$.

###### Proposition 3.7

[26, 33] Let $k\geq 3$ be an odd integer and let $A$ be a cyclically
$k$-diagonal Heffter array of size $n\geq k$. If $\gcd(n,k-1)=1$, then there
exist two compatible orderings $\omega_{r}$ and $\omega_{c}$ of the rows and
the columns of $A$.

###### Proposition 3.8

[26] Let $k\geq 3$ and let $A$ be a cyclically $k$-diagonal Heffter array of
size $n\geq(k-2)(k-1)$. Then there exist two compatible orderings $\omega_{r}$
and $\omega_{c}$ of the rows and the columns of $A$ if and only if $n$ and $k$
are both odd.

In the case of an $H(m,n;h,k)$ with no empty cells (an $H(m,n)$) the following
theorem was proven in [39]. In the proof, an explicit compatible ordering of
the rows and columns is given.

###### Proposition 3.9

[39] Let $A$ be a $m\times n$ Heffter array with no empty cells where at least
one of $m$ and $n$ is odd. Then there exist compatible orderings, $\omega_{r}$
and $\omega_{c}$ on the rows and columns of $A$.

So from Theorem 3.5 and Proposition 3.9 we have necesary and sufficient
condition for an $H(m,n)$ to have compatible orderings. Specifically we have
the following.

###### Theorem 3.10

Let $A$ be a $m\times n$ Heffter array with no empty cells (an $H(m,n))$. Then
there exist compatible orderings, $\omega_{r}$ and $\omega_{c}$ on the rows
and columns of $A$ if and only if at least one of $m$ and $n$ is odd.

Compatible orderings are the link between Heffter arrays and biembeddings,
while the simplicity of a Heffter array will guarantee the biembedding is
simple also. The following fundamental theorem from [7] gives the connection
between Heffter arrays with compatible orderings and biembeddings. The
interested reader is referred to that paper for the full details of this
connection, but briefly, using the compatible orderings of the Heffter array
one obtains a current graph which in turn gives the rotation of the edges
around each vertex. The rows and columns give the cycles in the two cycle
decompositions.

###### Theorem 3.11

[7] Given a Heffter array $H(m,n;h,k)$ with compatible orderings $\omega_{r}$
on the rows and $\omega_{c}$ on the columns, there exists a biembedding of the
complete graph $K_{2nk+1}$ on an orientable surface such that every edge is on
a face of size $h$ and a face of size $k$. Moreover, if $\omega_{r}$ and
$\omega_{c}$ are both simple, then all faces are simple cycles.

A $k$-cycle system of order $n$ is a set of cycles of length $k$, denoted by
$C_{k}$, whose edges partition the edges of $K_{n}$, one speaks also of a
$C_{k}$-decomposition of $K_{n}$. Restating the previous theorem in terms of
biembeddings of cycle systems we have the following.

###### Theorem 3.12

[7] Given a Heffter array $H(m,n;h,k)$ with simple compatible orderings
$\omega_{r}$ on the rows and $\omega_{c}$ on the columns, there exists an
orientable biembedding of a $C_{h}$-decomposition and a $C_{k}$-decomposition
both of $K_{2nk+1}$, or equivalently, there is an orientable biembedding of an
$h$-cycle system and a $k$-cycle system both in $K_{2nk+1}$.

We note again that a discussion of the construction of cycle systems from
Heffter arrays will be given in Section 5. As an example of the application of
Proposition 3.9 and Theorems 3.11 and 3.12 simple $H(3,n)$ were explicitly
constructed in [39] for all $n\geq 3$ with an eye towards the biembedding
problem. The following is the main result in that paper.

###### Theorem 3.13

[39] There exists a globally simple $H(3,n)$ for every $n\geq 3$. Thus for
every $n\geq 3$ there exists a biembedding of $K_{6n+1}$ on an orientable
surface such that each edge is on a $3$-cycle and an $n$-cycle and hence there
is a biembedding of a Steiner triple system and an $n$-cycle system.

The following problem was introduced in [26] in view of its importance in
looking for compatible orderings for Heffter arrays. However, it can also be
considered as a tour problem on a toroidal board which is interesting _per
se_.

Let $A$ be an $m\times n$ toroidal partially filled array. By $r_{i}$ we
denote the orientation of the $i$-th row of $A$, precisely $r_{i}=1$ if it is
from left to right and $r_{i}=-1$ if it is from right to left. Analogously,
for the $j$-th column, if its orientation $c_{j}$ is from top to bottom then
$c_{j}=1$ otherwise $c_{j}=-1$. Assume that an orientation
$\mathcal{R}=(r_{1},\dots,r_{m})$ and $\mathcal{C}=(c_{1},\dots,c_{n})$ is
fixed. Given an initial filled cell $(i_{1},j_{1})$ consider the sequence
$L_{\mathcal{R},\mathcal{C}}(i_{1},j_{1})=((i_{1},j_{1}),(i_{2},j_{2}),\ldots,(i_{\ell},j_{\ell}),$
$(i_{\ell+1},j_{\ell+1}),\ldots)$ where $j_{\ell+1}$ is the column index of
the filled cell $(i_{\ell},j_{\ell+1})$ of the row $R_{i_{\ell}}$ next to
$(i_{\ell},j_{\ell})$ in the orientation $r_{i_{\ell}}$, and where
$i_{\ell+1}$ is the row index of the filled cell of the column
$C_{j_{\ell+1}}$ next to $(i_{\ell},j_{\ell+1})$ in the orientation
$c_{j_{\ell+1}}$. The problem is the following:

Crazy Knight’s Tour Problem.[26] Given a toroidal partially filled array $A$,
do there exist $\mathcal{R}$ and $\mathcal{C}$ such that the list
$L_{\mathcal{R},\mathcal{C}}$ covers all the filled cells of $A$?

The Crazy Knight’s Tour Problem for a given array $A$ is denoted by $P(A)$.
Also, given a filled cell $(i,j)$, if $L_{\mathcal{R},\mathcal{C}}(i,j)$
covers all the filled positions of $A$, then $(\mathcal{R},\mathcal{C})$ is
said to be a solution of $P(A)$.

###### Example 3.14

Let $A$ be the following array, where the bullets “$\bullet$” denote the
filled cells:

$\begin{array}[]{|c|c|c|c|}\hline\cr\bullet&&&\bullet\\\
\hline\cr&\bullet&\bullet&\\\ \hline\cr&\bullet&\bullet&\bullet\\\
\hline\cr\bullet&&\bullet&\\\ \hline\cr\end{array}$

Choosing $\mathcal{R}:=(-1,1,1,-1)$ and $\mathcal{C}:=(1,-1,1,1)$ we can cover
all the filled positions of $A$, as shown in the table below where in each
filled cell we write $j$ if this position is the $j$-th element of the list
$L_{\mathcal{R},\mathcal{C}}(1,1)$. The elements of $\mathcal{R}$ and
$\mathcal{C}$ are represented by an arrow.

$\begin{array}[]{r|r|r|r|r|}&\downarrow&\uparrow&\downarrow&\downarrow\\\
\hline\cr\leftarrow&1&&&5\\\ \hline\cr\rightarrow&&3&7&\\\
\hline\cr\rightarrow&&8&4&2\\\ \hline\cr\leftarrow&6&&9&\\\
\hline\cr\end{array}$

###### Remark 3.15

If $A$ is an $H(m,n;h,k)$ such that $P(A)$ admits a solution
$(\mathcal{R},\mathcal{C})$, then $A$ admits two compatible orderings
$\omega_{r}$ and $\omega_{c}$ that can be determined as follows. For each row
$R$ (column $C$, respectively) of $A$, let $\omega_{R}$ be the natural
ordering if $r_{i}=1$ ($c_{i}=1$, respectively) and its inverse otherwise.
Setting $\omega_{r}=\omega_{R_{1}}\circ\ldots\circ\omega_{R_{m}}$ and
$\omega_{c}=\omega_{C_{1}}\circ\ldots\circ\omega_{C_{n}}$, since
$(\mathcal{R},\mathcal{C})$ is a solution of $P(A)$, then $\omega_{r}$ and
$\omega_{c}$ are compatible.

The relationship between globally simple Heffter arrays, Crazy Knight’s Tour
Problem and biembeddings is explained in the following result. It is a
reformulation of Theorem 3.11 in the case of square globally simple Heffter
arrays in terms of $P(A)$. Clearly, a similar theorem holds in the rectangular
case.

###### Theorem 3.16

Let $A$ be a globally simple Heffter array $H(n;k).$ If there exists a
solution of $P(A)$, then there exists a biembedding of $K_{2nk+1}$ on an
orientable surface whose face boundaries are $k$-cycles.

The existence of integer globally simple Heffter arrays $H(n;k)$ with
compatible orderings for $n\equiv 1\pmod{4}$ and $k\equiv 3\pmod{4}$ has been
proved (with infinite sporadic exceptions) in [23].

###### Theorem 3.17

[23] Let $n\equiv 1\pmod{4}$, $t>0$ and $n>4t+3$. If there exists $\alpha$
such that $2t+2\leq\alpha\leq n-2-2t$, $\gcd(n,\alpha)=1$,
$\gcd(n,\alpha-2t-1)=1$ and $\gcd(n,n-1-\alpha-2t)=1$, then there exists a
globally simple integer Heffter array $H(n;4t+3)$ with row and column
orderings that are both simple and compatible.

###### Theorem 3.18

[23] Let $n\equiv 1\pmod{4}$, $n>k\equiv 3\pmod{4}$ and either: $(a)$ $n$ is
prime; $(b)$ $n=k+2$; or $(c)$ $n\geq 7(k+1)/3$ and if $n\equiv 3\pmod{6}$
then $k\equiv 7\pmod{12}$. Then there exists a globally simple integer Heffter
array $H(n;k)$ with an ordering that is both simple and compatible.
Furthermore [by Theorem 3.12], there exists a face $2$-colorable embedding
$\mathcal{G}$ of $K_{2nk+1}$ on an orientable surface such that the faces of
each color are cycles of length $k$. Moreover, $\mathbb{Z}_{2nk+1}$ has a
sharply vertex-transitive action on $\mathcal{G}$.

###### Definition 3.19

Two Heffter arrays $H$ and $H^{\prime}$ are said to be equivalent if one can
be obtained from the other by (i) rearranging rows or columns; (ii) replacing
every entry $x$ in $H$ with $-x$; and/or (iii) taking the transpose.

The following theorem gives a lower bound on the number of inequivalent
Heffter arrays $H(n;4t+3)$ that satisfy Theorem 3.17. Let $\mathcal{H}(n)$
represent the number of derangements on $\\{0,1,\ldots,n-1\\}$.

###### Theorem 3.20

[23] Let $n\equiv 1\pmod{4}$, $t\geq 2$ and $n>4t+3$. If there exists $\alpha$
such that $2t+2\leq\alpha\leq n-2-2t$, $\gcd(n,\alpha)=1$,
$\gcd(n,\alpha-2t-1)=1$ and $\gcd(n,n-1-\alpha-2t)=1$, then there exist at
least $(n-2)(\mathcal{H}(t-2))^{2}\simeq(n-2)[(t-2)!/e]^{2}$ inequivalent
globally simple integer Heffter arrays $H(n;4t+3)$, each with an ordering that
is both simple and compatible.

The other case, that is $n\equiv 3\pmod{4}$ and $k\equiv 1\pmod{4}$, remains
unsolved in general. In the same paper the authors also investigated the
number of biembeddings of $K_{v}$ obtained from certain classes of Heffter
arrays. The following result is a corollary of Theorem 3.20.

###### Theorem 3.21

[23] Let $n\equiv 1\pmod{4}$ and $k\equiv 3\pmod{4}$, $k\geq 11$ and either
$n$ is prime or $n\geq(7k+1)/3$. Further if $n\equiv 3\pmod{6}$ assume
$k\equiv 7\pmod{12}$. Then there exists at least $(n-2)[((k-11)/4)!/e]^{2}$
distinct face $2$-colorable embeddings of the complete graph $K_{2nk+1}$ onto
an orientable surface where each face is a cycle of fixed length $k$, and the
vertices can be labeled with the elements of $\mathbb{Z}_{2nk+1}$ in such a
way that this group $(\mathbb{Z}_{2nk+1},+)$ has a sharply vertex-transitive
action on the embedding.

Starting from the results of [23], in [36] the authors consider the number of
non-isomorphic biembeddings of $K_{v}$ obtained from certain classes of
Heffter arrays. Set the binary entropy function by
$H(p):=-p\log_{2}{p}-(1-p)\log_{2}(1-p)$.

###### Theorem 3.22

[36] Let $v=2nk+1$, $k=4t+3$ and let $n\equiv 1\pmod{4}$ be either a prime or
$n\geq(7k+1)/3$. Moreover, if $n\equiv 3\pmod{6}$ assume $k\equiv 7\pmod{12}$.
Then, the number of non-isomorphic face $2$-colorable embeddings of the
complete graph $K_{2nk+1}$ onto an orientable surface where each face is a
cycle of fixed length $k$ is at least

$\frac{(n-2)[\mathcal{H}(t-2)]^{2}}{2(2nk)^{2}}\approx\frac{\pi(t-2)^{2t-5}}{64e^{2t-2}n}\approx
k^{k/2+o(k)}/v.$

###### Theorem 3.23

[36] Let $k\geq 3$ be odd, $n\geq 4k-3$ and $nk\equiv 3\pmod{4}$. Assume also
that $\gcd(n,k-1)=1$. Then, set $v=2nk+1$, the number of non-isomorphic face
$2$-colorable embeddings of the complete graph $K_{2nk+1}$ onto an orientable
surface where each face has fixed length $k$ is at least:

$\frac{\binom{\lceil n/(k-1)\rceil}{\lceil
n/(4k-4)\rceil}}{(nk)^{2}}\gtrsim\frac{\sqrt{\frac{2(k-1)}{3n\pi}}2^{\frac{n}{k-1}\cdot
H(1/4)+1}}{(nk)^{2}}\approx 2^{v\cdot\frac{H(1/4)}{2k(k-1)}+o(v,k)}.$

Moreover, if:

  * $(\rm{1})$

$k=3$ the number of _simple_ non-isomorphic face $2$-colorable embeddings of
$K_{6n+1}$ is at least $\frac{2^{n/2}}{9n^{2}}\approx 2^{v/12+o(v)}$;

  * $(\rm{2})$

$k=5,7,9$ the number of _simple_ non-isomorphic face $2$-colorable embeddings
of $K_{2nk+1}$ is at least $\frac{\binom{\lfloor n/(k-1)\rfloor}{\lfloor
n/(4k-4)\rfloor}}{(nk)^{2}}\gtrsim\frac{\sqrt{\frac{2(k-1)}{3n\pi}}2^{\lfloor\frac{n}{k-1}\rfloor\cdot
H(1/4)+1}}{(nk)^{2}}\approx 2^{v\cdot\frac{H(1/4)}{2k(k-1)}+o(v,k)}.$

One additional embedding result follows directly by Theorem 2.5.

###### Theorem 3.24

[33] There exists a biembedding of the complete graph of order $2nk+1$ and one
of the cocktail graph of order $2nk+2$ on orientable surfaces such that every
face is a $k$-cycle, whenever $k\in\\{3,5,7,9\\}$, $nk\equiv 3\pmod{4}$ and
$n>k$.

##  4 Sequencing Subsets of a Group

We know from Theorem 3.11 that if there is a simple Heffter array $H(m,n;h,k)$
(with compatible orderings), then there is a biembedding of $K_{2nk+1}$ on an
orientable surface such that every edge is on a face of size $h$ and a face of
size $k$ and all faces are simple cycles. It is thus a natural question to ask
just when the rows and columns of a Heffter array have a simple ordering, or
more generally, when subsets of a group have a simple ordering. Indeed, if
this is always the case, then Heffter arrays would always be simple and the
condition for simplicity of the array would be unnecessary. It has indeed been
an active area of research to show this, but at this time it is still not
known for all subsets (and all groups). In this section we give some
conjectures relating to the sequencing of subsets of groups as well as some of
the results in this area.

We begin with the more classical notion of a sequenceable group. Let $G$ be a
group of order $n$ (written additively). Given an ordering
$\omega=(0,g_{1},g_{2},\ldots,g_{n-1})$ of the elements in $G$, let
$s_{i}=\sum_{j=1}^{i}g_{j}$ be the $i$-th partial sum of $\omega$. The group
$G$ is sequenceable if there exists some ordering of the elements of $G$ where
all of these partial sums $s_{0}=0,s_{1},\ldots,s_{n-1}$ are distinct
(equivalently, if $G$ admits a simple ordering), while it is R-sequenceable if
$s_{0}=0,s_{1},\ldots,s_{n-2}$ are distinct and in addition $s_{n-1}=0$. The
concept of sequenceable group was introduced in 1961 by Gordon [47], even if
similar ideas for cyclic groups were already presented in 1892 by Lucas [58].
Alspach et al. [4] proved that any finite abelian group is either sequenceable
or R-sequenceable, confirming the Friedlander-Gordon-Miller conjecture.

Here is a short summary of the results concerning the sequencing and
R-sequencing of groups. For an extensive survey on this topic the reader can
refer to [66].

###### Proposition 4.1

Results on sequencing of groups.

  1. $(\rm{1})$

An abelian group is sequenceable if and only if it has a unique element of
order $2$.

  2. $(\rm{2})$

No non-abelian group of order less than $10$ is sequenceable. But Keedwell
conjectures that all non-abelian groups of order greater than $10$ are
sequenceable.

  3. $(\rm{3})$

Keedwell’s conjecture has been proved true for the following classes of
groups:

    1. $(\rm{a})$

All non-abelian groups of order $n$, $10\leq n\leq 32$;

    2. $(\rm{b})$

The dihedral groups $D_{n}$;

    3. $(\rm{c})$

$A_{5}$ and $S_{5}$;

    4. $(\rm{d})$

Solvable groups with a unique element of order $2$;

    5. $(\rm{e})$

Some non-solvable groups with a unique element of order $2$;

    6. $(\rm{f})$

Some groups of order $pq$, $p$ and $q$ odd primes.

###### Proposition 4.2

Results on R-sequencing of groups.

  1. $(\rm{1})$

If $G$ is an R-sequenceable group, then its Sylow $2$-subgroup is either
trivial or non-cyclic.

  2. $(\rm{2})$

The following groups are R-sequenceable:

    1. $(\rm{a})$

$\mathbb{Z}_{n}$, $n$ odd;

    2. $(\rm{b})$

Abelian groups of order $n$, $gcd(n,6)=1$;

    3. $(\rm{c})$

$D_{n}$, $n$ even;

    4. $(\rm{d})$

$Q_{2n}$ if $n+1$ is a prime of the form $4k+1$, for which $-2$ is a primitive
root;

    5. $(\rm{e})$

Non-abelian groups of order $pq$, $p<q$ odd primes, with $2$ a primitive root
of $p$.

The problem of ordering the elements of a subset of a given group in such a
way that all the partial sums are pairwise distinct is more recent than the
sequenceable group question. Nonetheless it has also been extensively studied,
most recently because of its connection to Heffter arrays, but not only for
this reason. Indeed there are several open conjectures on partial sums of a
finite group.

For example, several years ago Alspach made the following conjecture, whose
validity would shorten some cases of known proofs about the existence of cycle
decompositions.

###### Conjecture 4.3

Let $T\subseteq\mathbb{Z}_{v}\setminus\\{0\\}$ such that $\sum_{t\in T}t\neq
0$. Then there exists an ordering of the elements of $T$ such that the partial
sums are all distinct and non-zero.

The first published paper on this problem is by Bode and Harborth [11] in
which the authors proved that Conjecture 4.3 is valid if $|T|=v-1$ or
$|T|=v-2$ or if $v\leq 16$ (the latter was obtained by computer verification).
Recently, several other papers on this topic have been published, see [3, 27,
29, 38, 50, 67].

Another conjecture, very close to the Alspach’s conjecture, was originally
proposed in 1971 by Graham [44] for cyclic groups of prime order and then
extended in 2015 by Archdeacon et al. [10] to every cyclic group as follows.

###### Conjecture 4.4

[10] Let $T\subseteq\mathbb{Z}_{v}\setminus\\{0\\}$. Then $T$ admits a simple
ordering.

Clearly Conjecture 4.4 implies Conjecture 4.3. In [10] the authors proved that
Conjecture 4.3 implies Conjecture 4.4. Then they proved that Conjecture 4.4 is
valid if $|T|\leq 6$ and (by computer verification) for $v\leq 25$.

In [32], Costa et al. proposed the following conjecture on partial sums which
was also motivated by the study of Heffter arrays.

###### Conjecture 4.5

[32] Let $(G,+)$ be an abelian group. Let $T$ be a finite subset of
$G\setminus\\{0\\}$ such that no $2$-subset $\\{x,-x\\}$ is contained in $T$
and with the property that $\sum_{t\in T}t=0$. Then $T$ admits a simple
ordering.

If $G=\mathbb{Z}_{v}$, then every row and column of a Heffter array can be
viewed as a subset $T$ considered in Conjecture 4.5. Hence if this conjecture
were true for cyclic groups, then every Heffter array would be simple. In [32]
the validity of Conjecture 4.5 is proved theoretically for subsets $T$ of size
less than $10$ and, with the aid of a computer, for every abelian group $G$
with $|G|\leq 27$.

Clearly if $G=\mathbb{Z}_{v}$, Conjecture 4.5 immediately follows from
Conjecture 4.4. Indeed in [32] the authors extended Conjecture 4.4 to every
abelian group and proved by computer that this extended conjecture is valid
for every abelian group of order at most $23$. Also, they pointed out that in
[10], when proving the conjecture true for $|T|\leq 6$ the authors do not use
the hypothesis that $T$ is a subset of a cyclic group. In fact, these proofs
for $|T|\leq 6$ work in general for an arbitrary abelian group.

Some results are known about Conjectures 4.3 and 4.4 on an arbitrary group.
For instance the validity of Conjecture 4.3 has been verified by computer for
_all abelian groups_ of order at most $21$ in [32]. It has also been verified
in [3] for any subset $T$ of size at most $9$ of an _arbitrary abelian group_.
On the other hand, Conjecture 4.3 cannot be generalized to nonabelian groups
as shown in [6] by Anderson. Also, if we consider Conjectures 4.3 and 4.4 in
the case of abelian groups then, again, Conjecture 4.3 implies Conjecture 4.4.
Hence it follows that Conjecture 4.4 holds, in the abelian case, for any
subset $T$ of size at most $9$ and when $\sum_{t\in T}t=0$ for $T$ of size at
most $10$, for details see [10]. This implies that also Conjecture 4.5 holds
for subsets of size at most $10$. For Conjecture 4.4 it is also natural to
investigate the nonabelian case. In [32] this conjecture is proved
theoretically for subsets $T$ of size at most $5$ of _every arbitrary group_
and a computer verification is made for _all groups_ of order not exceeding
$19$.

Finally, we recall that a subset of an (additive) abelian group is said to be
_sequenceable_ if there is an ordering $(t_{1},\ldots,t_{k})$ of its elements
such that the partial sums $(s_{0},s_{1},\ldots,s_{k})$, where $s_{0}=0$, are
distinct, with the possible exception that we may have $s_{k}=s_{0}=0$. The
following theorem summarizes what is known about sequencing subsets of the
cyclic group. We should note that most of the recent progress on this problem
has been made by use of the polynomial method. This method relies on the non-
vanishing corollary to the Combinatorial Nullstellensatz (see [2, 27, 29, 50,
67]).

###### Theorem 4.6

Let $T\subseteq\mathbb{Z}_{v}\setminus\\{0\\}$ with $|T|=k$. Then $T$ is
sequenceable in the following cases:

  * $(\rm{1})$

$k\leq 9$ [3];

  * $(\rm{2})$

$k\leq 12$ when $v=pq$ where $p$ is prime and $q\leq 4$ [29, 50];

  * $(\rm{3})$

$k\leq 12$ when $v=mq$ where all the prime factors of $m$ are bigger than
$\frac{k!}{2}$ and $q\leq 4$ [29];

  * $(\rm{4})$

$k=v-3$ when $v$ is prime and $\sum_{t\in T}t=0$ [50];

  * $(\rm{5})$

$k=v-2$ when $\sum_{t\in T}t\neq 0$ [11];

  * $(\rm{6})$

$k=v-1$ [4, 47];

  * $(\rm{7})$

$v\leq 21$ and $v\leq 25$ when $\sum_{t\in T}t=0$ [10, 32].

##  5 Connections with difference families and graph decompositions

In this section we discuss the connection between Heffter arrays and cycle
decompositions of the complete graph. We have already discussed the connection
between Heffter arrays and biembeddings of two cycle systems (cycle
decompositions). In this section we will explicitly show how one can produce
the two cycle decompositions $\mathcal{D}$ and $\mathcal{D^{\prime}}$ and that
these cycle decompositions have the property that if two edges are in the same
cycle of $\mathcal{D}$, then they will be in different cycles in
$\mathcal{D}^{\prime}$.

First of all we recall some basic definitions about graphs and graph
decompositions. Good references about these concepts are [70] and [12],
respectively. Given a graph $\Gamma$, by $V(\Gamma)$ and $E(\Gamma)$ we mean
the vertex set and the edge set of $\Gamma$, respectively, and by
${}^{\lambda}\Gamma$ the multigraph obtained from $\Gamma$ by repeating each
edge $\lambda$ times. We denote by $K_{q\times r}$ the complete multipartite
graph with $q$ parts each of size $r$, obviously $K_{q\times 1}$ is nothing
but the complete graph $K_{q}$. Finally, by $P_{k}$ we mean the path of length
$k$.

Let $\Gamma$ be a subgraph of a graph $K$. A $\Gamma$-_decomposition_ of $K$
is a set $\mathcal{D}$ of subgraphs of $K$ isomorphic to $\Gamma$, called
_blocks_ , whose edges partition $E(K)$. As already remarked in the previous
section, a $C_{k}$-decomposition of $K_{v}$ is also called a $k$-_cycle system
of order_ $v$.

###### Definition 5.1

Two graph decompositions $\mathcal{D}$ and $\mathcal{D}^{\prime}$ of the same
graph $K$ are _orthogonal_ if for every $B\in\mathcal{D}$ and every
$B^{\prime}\in\mathcal{D}^{\prime}$, $B$ intersects $B^{\prime}$ in at most
one edge.

We note the following obvious fact connecting the notions of biembedded cycle
systems to orthogonal cycle systems. From this fact, many of our earlier
results on biembeddings of cycle systems give results on orthogonal cycle
systems.

###### Proposition 5.2

If there is a simple biembedding of a cycle system $\mathcal{D}$ and a cycle
system $\mathcal{D}^{\prime}$ in $K_{v}$, then $\mathcal{D}$ and
$\mathcal{D^{\prime}}$ are orthogonal cycle systems of order $v$.

If the vertices of $\Gamma$ belong to a group $G$, given $g\in G$, by
$\Gamma+g$ one means the graph whose vertex set is $V(\Gamma)+g$ and whose
edge set is $\\{\\{x+g,y+g\\}\mid\\{x,y\\}\in E(\Gamma)\\}$. An _automorphism
group_ of a $\Gamma$-decomposition $\mathcal{D}$ of $K$ is a group of
bijections on $V(K)$ leaving $\mathcal{D}$ invariant. A $\Gamma$-decomposition
of $K$ is said to be _regular under a group_ $G$ or $G$-_regular_ if it admits
$G$ as an automorphism group acting sharply transitively on $V(K)$. Heffter
arrays are related to _cyclic_ decompositions, namely decompositions regular
under a cyclic group. We recall the following result.

###### Proposition 5.3

Let $K$ be a graph with $v$ vertices. A $\Gamma$-decomposition $\mathcal{D}$
of $K$ is cyclic if and only if, up to isomorphisms, the following conditions
hold:

  * $(\rm{1})$

the set of vertices of $K$ is $\mathbb{Z}_{v}$;

  * $(\rm{2})$

for all $B\in\mathcal{D}$, $B+1\in\mathcal{D}$.

Clearly, to describe a cyclic decomposition it is sufficient to exhibit a
complete system $\mathcal{B}$ of representatives for the orbits of
$\mathcal{D}$ under the action of $\mathbb{Z}_{v}$. The elements of
$\mathcal{B}$ are called _base blocks_ of $\mathcal{D}$.

One of most efficient tools applied for constructing regular decompositions is
the _difference method_ , see [1]. In particular, difference families over
graphs have been introduced by Buratti [14], see also [17]. Given a graph
$\Gamma$ with vertices in an additive group $G$, the _list of differences_ of
$\Gamma$ is the multiset $\Delta\Gamma$ of all possible differences $x-y$
between two adjacent vertices of $\Gamma$. So, denoting by $D(\Gamma)$ the set
of _di-edges_ of $\Gamma$, i.e., the set of all ordered pairs $(x,y)$ with
$\\{x,y\\}\in E(\Gamma)$, one can write:

$\Delta\Gamma=\\{x-y\mid(x,y)\in D(\Gamma)\\}.$

More generally, if
$\mathcal{F}=\\{\Gamma_{1},\Gamma_{2},\ldots,\Gamma_{\ell}\\}$ is a collection
of graphs with vertices in $G$, one sets
$\Delta\mathcal{F}=\Delta\Gamma_{1}\cup\Delta\Gamma_{2}\cup\ldots\cup\Delta\Gamma_{\ell}$,
where in the union each element has to be counted with its multiplicity.

###### Definition 5.4

Let $J$ be a subgroup of a group $G$ and let $\Gamma$ be a graph. A collection
$\mathcal{F}$ of graphs isomorphic to $\Gamma$ and with vertices in $G$ is
said to be a $(G,J,\Gamma,\lambda)$-_difference family_ (briefly DF) _over $G$
relative to $J$_ if each element of $G\setminus J$ appears exactly $\lambda$
times in $\Delta\mathcal{F}$ while no element of $J$ appears there.

In the case of $J=\\{0\\}$ one simply says that $\mathcal{F}$ is a
$(G,\Gamma,\lambda)$-DF. Also, if $t$ is a divisor of $v$, by writing
$(v,t,\Gamma,\lambda)$-DF one means a
$(\mathbb{Z}_{v},\frac{v}{t}\mathbb{Z}_{v},\Gamma,\lambda)$-DF where
$\frac{v}{t}\mathbb{Z}_{v}$ denotes the subgroup of $\mathbb{Z}_{v}$ of order
$t$. Analogously, $(v,\Gamma,\lambda)$-DF means
$(\mathbb{Z}_{v},\Gamma,\lambda)$-DF. The connection between relative
difference families and decompositions of a complete multipartite multigraph
is given by the following result.

###### Proposition 5.5

[17] If $\mathcal{F}=\\{\Gamma_{1},\Gamma_{2},\ldots,\Gamma_{\ell}\\}$ is a
$(G,J,\Gamma,\lambda)$-DF, then $\mathcal{D}=\\{\Gamma_{i}+g\mid
i=1,\ldots,\ell;g\in G\\}$ is a $G$-regular $\Gamma$-decomposition of
${}^{\lambda}K_{q\times r}$, where $q=|G:J|$ and $r=|J|$. In particular, if
$\mathcal{F}$ is $(v,\Gamma,1)$-DF then $\mathcal{D}$ is a cyclic
$\Gamma$-decomposition of $K_{v}$.

Now we show how starting from a simple Heffter array one can construct a pair
of orthogonal cyclic cycle decompositions of the complete graph $K_{v}$ or of
the cocktail party graph $K_{2\ell}-I$, namely the complete graph $K_{2\ell}$
minus the $1$-factor $I$ whose edges are
$[0,\ell],[1,\ell+1],[2,\ell+2],\ldots,[\ell-1,2\ell-1]$. It is well known
that the problem of finding necessary and sufficient conditions for cyclic
cycle decompositions of $K_{v}$ and $K_{2\ell}-I$ has attracted much attention
(see, for instance, [15, 69, 71] and [13, 16, 51, 52], respectively).

First of all, from a simple Heffter array one can construct two cyclic
difference families whose blocks are cycles, as follows. Let $H$ be a simple
$H(m,n;h,k)$, hence every row and every column of $H$ admits a simple
ordering. Starting from a row of $H$ one can construct a cycle of length $h$
whose vertices are the partial sums of the simple ordering of such a row.
Denoted by $\Gamma_{i}$ the cycle obtained from the $i$-th row $R_{i}$ of $H$,
it is immediate to see that $\Delta\Gamma_{i}=\pm R_{i}$. Now, let
$\mathcal{F}_{R}$ be the set of the $m$ cycles of length $h$ constructed from
the $m$ rows of $H$ with this procedure. By Condition (b) of Definition 1.4,
we have $\Delta{\mathcal{F}_{R}}=\mathbb{Z}_{2mh+1}\setminus\\{0\\}$. Hence
$\mathcal{F}_{R}$ is a $(2mh+1,C_{h},1)$-DF. Clearly, with the same technique,
one can construct $n$ cycles of length $k$ starting from the columns of $H$.
Let $\Gamma^{\prime}_{j}$ be the cycle obtained from the $j$-th column and let
$\mathcal{F}_{C}$ be the set of these cycles. Again by Condition (b) of
Definition 1.4, we have
$\Delta{\mathcal{F}_{C}}=\mathbb{Z}_{2nk+1}\setminus\\{0\\}$. Hence
$\mathcal{F}_{C}$ is a $(2nk+1,C_{k},1)$-DF.

###### Remark 5.6

For every $\Gamma_{i}\in\mathcal{F}_{R}$ and every
$\Gamma^{\prime}_{j}\in\mathcal{F}_{C}$, we see that $\Gamma_{i}$ and
$\Gamma^{\prime}_{j}$ contain at most one edge in common and hence
$|\Delta\Gamma_{i}\cap\Delta\Gamma^{\prime}_{j}|\in\\{0,2\\}$. In particular,
the intersection is empty if the cell on the $i$-th row and the $j$-th column
of $H$ is empty, while the intersection has size two otherwise.

By Proposition 5.5, the cycles of $\mathcal{F}_{R}$ are the base blocks of a
cyclic $C_{h}$-decomposition $\mathcal{D}_{R}$ of $K_{2mh+1}$, while the
cycles of $\mathcal{F}_{C}$ are the base blocks of a cyclic
$C_{k}$-decomposition $\mathcal{D}_{C}$ of $K_{2nk+1}$. By Remark 5.6, the
decompositions $\mathcal{D}_{R}$ and $\mathcal{D}_{C}$ are orthogonal.

###### Example 5.7

Let $H$ be the $H(8;7)$ of Example 1.11 and let $\omega_{i}$’s and $\nu_{i}$’s
be the simple orderings given in Example 2.2. Considering the partial sums of
the $\omega_{i}$’s ($\nu_{i}$’s, respectively) in $\mathbb{Z}_{113}$ we obtain
the cycles $\Gamma_{i}$’s ($\Gamma^{\prime}_{i}$’s, respectively):

$\begin{array}[]{lcl}\Gamma_{1}=(8,33,49,22,-7,24,0);&&\Gamma^{\prime}_{1}=(8,47,30,-8,-51,-9,0);\\\
\Gamma_{2}=(-17,-23,-51,-28,-2,30,0);&&\Gamma^{\prime}_{2}=(16,10,-35,-45,3,52,0);\\\
\Gamma_{3}=(39,29,24,57,72,37,0);&&\Gamma^{\prime}_{3}=(23,18,65,47,1,-50,0);\\\
\Gamma_{4}=(-38,-56,-49,-85,-74,-40,0);&&\Gamma^{\prime}_{4}=(25,-3,12,19,-34,-56,0);\\\
\Gamma_{5}=(-43,-88,-41,-63,-60,-19,0);&&\Gamma^{\prime}_{5}=(-27,-1,10,13,68,54,0);\\\
\Gamma_{6}=(42,90,44,30,32,-12,0);&&\Gamma^{\prime}_{6}=(-21,11,44,8,27,29,0);\\\
\Gamma_{7}=(20,-31,-84,-29,-50,-49,0);&&\Gamma^{\prime}_{7}=(-13,-43,-78,-44,-32,-31,0);\\\
\Gamma_{8}=(-52,-43,7,63,9,-4,0);&&\Gamma^{\prime}_{8}=(-37,-61,-21,20,-24,-4,0).\end{array}$

Then $\mathcal{F}_{R}=\\{\Gamma_{1},\ldots,\Gamma_{8}\\}$ and
$\mathcal{F}_{C}=\\{\Gamma^{\prime}_{1},\ldots,\Gamma^{\prime}_{8}\\}$ are two
sets of base cycles of a pair of orthogonal cyclic $C_{7}$-decompositions of
$K_{113}$.

From the arguments in this section we get the following results. Note the
similarity of Proposition 5.9 to Theorem 3.12, however we see that for the
application of Heffter arrays to orthogonal cycle decompositions it is not
required that the orderings of the rows and columns of the Heffter array be
compatible. For results on orthogonal cycle decompositions see [20] and the
references therein.

###### Proposition 5.8

If there exists a simple $H(m,n;h,k)$, then there exist a $(2mh+1,C_{h},1)$-DF
and a $(2nk+1,C_{k},1)$-DF.

###### Proposition 5.9

If there exists a simple $H(m,n;h,k)$, then there exist a cyclic
$C_{h}$-decomposition and a cyclic $C_{k}$-decomposition both of $K_{2nk+1}$,
and these decompositions are orthogonal.

Obviously if the array is globally simple one can immediately construct the
cycles starting from the rows and from the columns, without providing the
simple orderings. Also note that the elements of an $H(m,n;h,k)$ belong to
$\mathbb{Z}_{2nk+1}$, hence when we say that an ordering of a row or of a
column is simple, we mean that its partial sums are pairwise distinct modulo
$2nk+1$. In [33], the authors showed that it is useful to find orderings whose
partial sums are pairwise distinct both modulo $2nk+1$ and modulo $2nk+2$. To
this end they give the following definition.

###### Definition 5.10

An $SH^{*}(m,n;h,k)$ is a globally simple $H(m,n;h,k)$ such that the natural
ordering of each row and each column is simple also modulo $2nk+2$.

The arrays of Theorem 2.5 have this additional property. The usefulness of
these arrays is explained by the following proposition.

###### Proposition 5.11

[33] If there exists an $SH^{*}(m,n;h,k)$, then there exist:

  * (1)

a cyclic $C_{h}$-decomposition and a cyclic $C_{k}$-decomposition both of
$K_{2nk+1}$ and these decompositions are orthogonal;

  * (2)

a cyclic $C_{h}$-decomposition and a cyclic $C_{k}$-decomposition both of
$K_{2nk+2}-I$ and these decompositions are orthogonal.

###### Example 5.12

The following is an $SH^{*}(10;8)$ over $\mathbb{Z}_{161}$:

$\begin{array}[]{|r|r|r|r|r|r|r|r|r|r|}\hline\cr
77&80&-78&-71&-70&-79&&&69&72\\\ \hline\cr&&-17&-20&-25&-28&26&19&18&27\\\
\hline\cr 5&8&13&16&-14&-7&-6&-15&&\\\ \hline\cr
34&43&&&-33&-36&-41&-44&42&35\\\ \hline\cr&&21&24&29&32&-30&-23&-22&-31\\\
\hline\cr 58&51&50&59&&&-49&-52&-57&-60\\\
\hline\cr-38&-47&&&37&40&45&48&-46&-39\\\
\hline\cr-73&-76&74&67&66&75&&&-65&-68\\\
\hline\cr-62&-55&-54&-63&&&53&56&61&64\\\ \hline\cr-1&-4&-9&-12&10&3&2&11&&\\\
\hline\cr\end{array}$

By the partial sums in $\mathbb{Z}_{161}$ of the natural orderings of the rows
(columns, respectively) we obtain the cycles $\Gamma_{i}$’s
($\Gamma^{\prime}_{i}$’s, respectively):

$\begin{array}[]{lcl}\Gamma_{1}=(77,157,79,8,-62,-141,-72,0);&&\Gamma^{\prime}_{1}=(77,82,116,13,136,63,1,0);\\\
\Gamma_{2}=(-17,-37,-62,-90,-64,-45,-27,0);&&\Gamma^{\prime}_{2}=(80,88,131,21,135,59,4,0);\\\
\Gamma_{3}=(5,13,26,42,28,21,15,0);&&\Gamma^{\prime}_{3}=(-78,-95,-82,-61,-11,63,9,0);\\\
\Gamma_{4}=(34,77,44,8,-33,-77,-35,0);&&\Gamma^{\prime}_{4}=(-71,-91,-75,-51,8,75,12,0);\\\
\Gamma_{5}=(21,45,74,106,76,53,31,0);&&\Gamma^{\prime}_{5}=(-70,-95,-109,-142,-113,-76,-10,0);\\\
\Gamma_{6}=(58,109,159,57,8,117,60,0);&&\Gamma^{\prime}_{6}=(-79,-107,-114,-150,-118,-78,-3,0);\\\
\Gamma_{7}=(-38,-85,-48,-8,37,85,39,0);&&\Gamma^{\prime}_{7}=(26,20,-21,-51,-100,-55,-2,0);\\\
\Gamma_{8}=(-73,-149,-75,-8,58,133,68,0);&&\Gamma^{\prime}_{8}=(19,4,-40,-63,-115,-67,-11,0);\\\
\Gamma_{9}=(-62,-117,-10,-73,-20,-125,-64,0);&&\Gamma^{\prime}_{9}=(69,87,129,107,50,4,-61,0);\\\
\Gamma_{10}=(-1,-5,-14,-26,-16,-13,-11,0);&&\Gamma^{\prime}_{10}=(72,99,134,103,43,4,-64,0).\end{array}$

Then $\mathcal{F}_{R}=\\{\Gamma_{1},\ldots,\Gamma_{10}\\}$ and
$\mathcal{F}_{C}=\\{\Gamma^{\prime}_{1},\ldots,\Gamma^{\prime}_{10}\\}$ are
two sets of base cycles of a pair of orthogonal cyclic $C_{8}$-decompositions
of $K_{161}$.
Analogously, if we consider the partial sums of each row (column,
respectively) in $\mathbb{Z}_{162}$, we obtain the cycles
$\widetilde{\Gamma}_{i}$’s ($\widetilde{\Gamma}^{\prime}_{j}$’s, respectively)
where:

$\begin{array}[]{lcl}\widetilde{\Gamma}_{i}=\Gamma_{i},\quad i\neq
6,9;&&\widetilde{\Gamma}^{\prime}_{j}=\Gamma_{j},\quad j\neq 1,2;\\\
\widetilde{\Gamma}_{6}=(58,109,159,56,7,117,60,0);&&\widetilde{\Gamma}^{\prime}_{1}=(77,82,116,12,136,63,1,0);\\\
\widetilde{\Gamma}_{9}=(-62,-117,-9,-72,-19,-125,-64,0);&&\widetilde{\Gamma}^{\prime}_{2}=(80,88,131,20,135,59,4,0).\end{array}$

Now
$\widetilde{\mathcal{F}_{R}}=\\{\widetilde{\Gamma}_{1},\ldots,\widetilde{\Gamma}_{10}\\}$
and
$\widetilde{\mathcal{F}_{C}}=\\{\widetilde{\Gamma}^{\prime}_{1},\ldots,\widetilde{\Gamma}^{\prime}_{10}\\}$
are two sets of base cycles of a pair of orthogonal cyclic
$C_{8}$-decompositions of the cocktail party graph $K_{162}-I$.

##  6 Variants and generalizations

In this section we will discuss several of the variants and generalizations of
Heffter arrays that have been researched in earlier publications.

###  6.1 Weak Heffter arrays

In the original paper on Heffter arrays [7], Archdeacon also proposed the
following weaker type of Heffter array. These weaker Heffter arrays again have
applications to biembedding cycle systems as well as to orthogonal cycle
systems. Two Heffter systems $D_{h}=D(2mh+1,h)$ and $D_{k}=D(2nk+1,k)$ with
$mh=nk$ are weakly sub-orthogonal if the $i$-th part of $D_{h}$ has at most
one element $a_{i,j}$ such that either $a_{i,j}$ or $-a_{i,j}$ is in the
$j$-th part of $D_{k}$. Form a weak Heffter array $H(m,n;h,k)$ by placing
$a_{i,j}$ in the cell on the $i$-th row and the $j$-th column. The upper sign
on $\pm$ or $\mp$ is the row sign corresponding to its sign on $a_{i,j}$ in
$D_{h}$, the lower sign is the column sign used in $D_{k}$. Using the row
signs we get row sums 0 and the column signs give column sums 0.

The following is a formal definition of a weak Heffter array.

###### Definition 6.1

A _weak Heffter array_ $H(m,n;h,k)$ is an $m\times n$ matrix $A$ such that:

  * $(\rm{a_{1}})$

each row contains $h$ filled cells and each column contains $k$ filled cells;

  * $(\rm{b_{1}})$

for every $x\in\mathbb{Z}_{2nk+1}\setminus\\{0\\}$, there is exactly one cell
of $A$ whose element is one of the following: $x,-x,\pm x,\mp x$, where the
upper sign on $\pm$ or $\mp$ is the row sign and the lower sign is the column
sign;

  * $(\rm{c_{1}})$

the elements in every row and column (with the corresponding sign) sum to $0$
in $\mathbb{Z}_{2nk+1}$.

###### Example 6.2

Consider the following Heffter systems on a half-set of $\mathbb{Z}_{25}$:

$D_{4}=\\{\\{1,-7-6,12\\},\\{2,-4,10,-8\\},\\{-3,-11,9,5\\}\\}$

and

$D_{3}=\\{\\{1,2,-3\\},\\{-7,-4,11\\},\\{-6,-10,-9\\},\\{12,8,5\\}\\}.$

Note that they are weakly sub-orthogonal and starting from these two systems
one gets the following $3\times 4$ weak Heffter array over $\mathbb{Z}_{25}$
without empty cells :

$\begin{array}[]{|r|r|r|r|}\hline\cr 1&-7&-6&12\\\ \hline\cr 2&-4&\pm 10&\mp
8\\\ \hline\cr-3&\mp 11&\pm 9&5\\\ \hline\cr\end{array}$

To date there are no published papers specifically on weak Heffter arrays, but
one is in preparation, see [31]. In [7] Archdeacon showed how these arrays are
related to current graphs and hence to biembeddings (in this case on a non-
orientable surface). Also, by reasoning analogous to that in Section 5, it is
easy to see that the following holds.

###### Proposition 6.3

If there exists a simple weak $H(m,n;h,k)$, then there exist a
$(2mh+1,C_{h},1)$-DF and a $(2nk+1,C_{k},1)$-DF.

###### Proposition 6.4

If there exists a simple weak $H(m,n;h,k)$, then there exist a cyclic
$C_{h}$-decomposition and a cyclic $C_{k}$-decomposition both of $K_{2nk+1}$.
These decompositions are orthogonal.

###  6.2 $\lambda$-fold relative Heffter arrays

In [34, 35] the authors proposed a natural generalization of Heffter arrays,
which is related to signed magic arrays, difference families, graph
decompositions and biembeddings. In this generalization, the array is missing
the elements of a subgroup $J\subset\mathbb{Z}_{v}$ and each element of
$\mathbb{Z}_{v}\setminus J$ occurs exactly $\lambda$ times in the array. We
give the following definition.

###### Definition 6.5

Let $v=\frac{2nk}{\lambda}+t$ be a positive integer, where $t$ divides
$\frac{2nk}{\lambda}$, and let $J$ be the subgroup of $\mathbb{Z}_{v}$ of
order $t$. A $\lambda$-_fold Heffter array $A$ over $\mathbb{Z}_{v}$ relative
to $J$_, denoted by ${}^{\lambda}H_{t}(m,n;h,k)$, is an $m\times n$ array with
elements in $\mathbb{Z}_{v}\setminus J$ such that:

  * $(\rm{a_{2}})$

each row contains $h$ filled cells and each column contains $k$ filled cells;

  * $(\rm{b_{2}})$

the multiset $\\{\pm x\mid x\in A\\}$ contains each element of
$\mathbb{Z}_{v}\setminus J$ exactly $\lambda$ times;

  * $(\rm{c_{2}})$

the elements in every row and column sum to $0$ in $\mathbb{Z}_{v}$.

The terminology is due to their connection with _relative_ difference
families. Trivial necessary conditions for the existence of a
${}^{\lambda}H_{t}(m,n;h,k)$ are $mh=nk$, $2\leq h\leq n$ and $2\leq k\leq m$.
It is easy to see that condition $(\rm{b_{2}})$ of Definition 6.5 asks that
for every $x\in\mathbb{Z}_{v}\setminus J$ with $x$ different from the
involution the number of occurrences of $x$ and $-x$ in the array is
$\lambda$, while if the involution exists and it does not belong to $J$, then
it has to appear exactly $\frac{\lambda}{2}$ times, also no element of $J$
appears in the array.

If ${}^{\lambda}H_{t}(m,n;h,k)$ is a square array, then it is denoted by
${}^{\lambda}H_{t}(n;k)$. If $\lambda=t=1$ one finds again the classical
concept of Heffter array, i.e. an ${}^{1}H_{1}(m,n;h,k)$ is just an
$H(m,n;h,k)$. In general if $\lambda=1$, then it is omitted; the same holds
for $t$. Note also that if $\lambda=1$, then $3\leq h\leq n$ and $3\leq k\leq
m$.

###### Example 6.6

A ${}^{3}H_{2}(4;3)$ whose elements belong to $\mathbb{Z}_{10}$ and a
${}^{4}H_{4}(4;2)$ whose elements belong to $\mathbb{Z}_{8}$:
$\begin{array}[]{|r|r|r|r|}\hline\cr&1&2&-3\\\ \hline\cr 4&&4&2\\\
\hline\cr-3&2&&1\\\ \hline\cr-1&-3&4&\\\ \hline\cr\end{array}$
$\begin{array}[]{|r|r|r|r|}\hline\cr 1&-1&&\\\ \hline\cr-1&1&&\\\
\hline\cr&&3&-3\\\ \hline\cr&&-3&3\\\ \hline\cr\end{array}$

As was the case for Heffter arrays, a $\lambda$-fold relative Heffter array is
called _integer_ if condition ($\rm{c_{2}}$) in Definition 6.5 is strengthened
so that the elements in every row and every column, seen as integer in
$\pm\left\\{1,2,\ldots,\lfloor\frac{v}{2}\rfloor\right\\}$, sum to $0$ in
$\mathbb{Z}$. The ${}^{3}H_{2}(4;3)$ of Example 6.6 is not an integer Heffter
array, while the ${}^{4}H_{4}(4;2)$ in the same example is integer.

###### Example 6.7

An integer ${}^{2}H(5;3)$ over $\mathbb{Z}_{16}$ (note that the involution
appears exactly once in the array) and an integer ${}^{2}H(6;4)$ over
$\mathbb{Z}_{25}$:

$\begin{array}[]{|r|r|r|r|r|}\hline\cr 3&5&-8&&\\\ \hline\cr&2&5&-7&\\\
\hline\cr&&3&1&-4\\\ \hline\cr-4&&&6&-2\\\ \hline\cr 1&-7&&&6\\\
\hline\cr\end{array}$ $\begin{array}[]{|r|r|r|r|r|r|}\hline\cr&&1&-2&-5&6\\\
\hline\cr&&-3&4&7&-8\\\ \hline\cr-9&10&&&1&-2\\\ \hline\cr 11&-12&&&-3&4\\\
\hline\cr 5&-6&-9&10&&\\\ \hline\cr-7&8&11&-12&&\\\ \hline\cr\end{array}$

####  6.2.1 Necessary conditions

In [35] the authors established some necessary conditions for the existence of
a ${}^{\lambda}H_{t}(m,n;h,k)$. Firstly, it is trivial to note that Conditions
($\rm{a_{2}})$ and ($\rm{b_{2}})$ of Definition 6.5 imply that if there exists
a ${}^{\lambda}H_{t}(m,n;h,k)$ with either $h=2$ or $k=2$, then $\lambda$ has
to be even.

###### Proposition 6.8

[35] Suppose that there exists a ${}^{\lambda}H_{t}(m,n;h,k)$ and set
$v=\frac{2nk}{\lambda}+t$. If either $v$ is odd or $v$ and $t$ are even, then
$\lambda$ has to be a divisor of $nk$.

###### Proposition 6.9

[35] If $\lambda\equiv 2\pmod{4}$, $v=\frac{2nk}{\lambda}+t\equiv 2\pmod{4}$
and $t$ is odd, then a ${}^{\lambda}H_{t}(m,n;h,k)$ cannot exist.

About the integer case there are the following more restrictive conditions.

###### Proposition 6.10

[34, 35] Suppose that there exists an integer ${}^{\lambda}H_{t}(m,n;h,k)$
with $\lambda$ odd.

  * $(\rm{1})$

If $t$ divides $\frac{nk}{\lambda}$, then $\frac{nk}{\lambda}\equiv 0\pmod{4}$
or $\frac{nk}{\lambda}\equiv-t\equiv\pm 1\pmod{4}.$

  * $(\rm{2})$

If $t=\frac{2nk}{\lambda}$, then $h$ and $k$ must be even.

  * $(\rm{3})$

If $t\neq\frac{2nk}{\lambda}$ does not divide $\frac{nk}{\lambda}$, then
$\frac{2nk}{\lambda}+t\equiv 0\pmod{8}.$

It is known that these conditions are not sufficient. For instance in [34] it
is proved that there is no integer $H_{3n}(n;3)$ for $n\geq 3$ and no integer
$H_{8}(4;3)$, moreover in [31] it is shown the nonexistence of an integer
$H_{t}(4;3)$ for $t=4,6$.

####  6.2.2 Integer relative Heffter arrays with $\lambda=1$

Turning to the existence question now, the first class investigated is that of
integer relative Heffter arrays with $\lambda=1$.

###### Theorem 6.11

[34] Let $3\leq k\leq n$ with $k\neq 5$. There exists an integer $H_{k}(n;k)$
if and only if one of the following holds:

  * $(\rm{1})$

$k$ is odd and $n\equiv 0,3\pmod{4}$;

  * $(\rm{2})$

$k\equiv 2\pmod{4}$ and $n$ is even;

  * $(\rm{3})$

$k\equiv 0\pmod{4}$.

Furthermore, there exists an integer $H_{5}(n;5)$ if $n\equiv 3\pmod{4}$ and
it does not exist if $n\equiv 1,2\pmod{4}$.

The case $k=5$ and $n\equiv 0\pmod{4}$ is still open. For these values of the
parameters integer weak Heffter arrays have been constructed in [31].

###### Theorem 6.12

[37] There exists an integer globally simple $H_{t}(n;k)$ in the following
cases:

  * $(\rm{1})$

$k=3$, $t=n,2n$ for all odd $n$;

  * $(\rm{2})$

$t=k=3,5,7,9$ for all $n\equiv 3\pmod{4}$.

###### Theorem 6.13

[63] Let $m,n,h,k$ be such that $4\leq h\leq n$, $4\leq k\leq m$ and $mh=nk$.
Let $t$ be a divisor of $2nk$.

  * $(\rm{1})$

If $h,k\equiv 0\pmod{4}$, then there exists an integer $H_{t}(m,n;h,k)$.

  * $(\rm{2})$

If $h\equiv 2\pmod{4}$ and $k\equiv 0\pmod{4}$, then there exists an integer
$H_{t}(m,n;h,k)$ if and only if $m$ is even.

  * $(\rm{3})$

If $h\equiv 0\pmod{4}$ and $k\equiv 2\pmod{4}$, then there exists an integer
$H_{t}(m,n;h,k)$ if and only if $n$ is even.

  * $(\rm{4})$

Suppose that $m$ and $n$ are both even. If $h,k\equiv 2\pmod{4}$, then there
exists an integer $H_{t}(m,n;h,k)$.

###### Theorem 6.14

[65] Let $m,n,h,k$ be such that $3\leq h\leq n$, $3\leq k\leq m$ and $mh=nk$.
Suppose $d=\gcd(h,k)\geq 3$.

  * $(\rm{1})$

If $d\equiv 1\pmod{4}$ and $nk\equiv 3\pmod{4}$, then there exists an integer
$H_{d}(m,n;h,k)$.

  * $(\rm{2})$

If $d\equiv 3\pmod{4}$ and $nk\equiv 0,1\pmod{4}$, then there exists an
integer $H_{d}(m,n;h,k)$.

  * $(\rm{3})$

If $d=3$ and $nk$ is odd, then there exists an integer
$H_{\frac{nk}{3}}(m,n;h,k)$ and an integer $H_{\frac{2nk}{3}}(m,n;h,k)$.

####  6.2.3  $\lambda$-fold relative Heffter arrays

The following are the known existence results on $\lambda$-fold relative
Heffter arrays.

###### Theorem 6.15

[65] Let $m,n,h,k$ be such that $3\leq h\leq n$, $3\leq k\leq m$ and $mh=nk$.
Suppose $d=\gcd(h,k)\geq 3$.

  * $(\rm{1})$

If $d=3$ and $nk\equiv 3\pmod{4}$, then there exists a ${}^{2}H(m,n;h,k)$.

  * $(\rm{2})$

If $d=3$ and $nk\equiv 1\pmod{4}$, then there exists a ${}^{3}H(m,n;h,k).$

  * $(\rm{3})$

If $d=3$, $nk$ is odd, and $\lambda$ divides $n$, then there exists a
${}^{\lambda}H_{\frac{n}{\lambda}}(m,n;h,k).$

  * $(\rm{4})$

If $d=3$, $nk$ is odd, and $\lambda$ divides $2n$, then there exists a
${}^{\lambda}H_{\frac{2n}{\lambda}}(m,n;h,k).$

  * $(\rm{5})$

If $d=5$ and $nk\equiv 3\pmod{4}$, then there exists a ${}^{2}H(m,n;h,k).$

###### Theorem 6.16

[64] Let $m,n,h,k$ be such that $4\leq h\leq n$, $4\leq k\leq m$ and $mh=nk$.
Let $\lambda$ be a divisor of $2nk$ and let $t$ be a divisor of
$\frac{2nk}{\lambda}$. There exists an integer ${}^{\lambda}H_{t}(m,n;h,k)$ in
each of the following cases:

  * $(\rm{1})$

$h,k\equiv 0\pmod{4}$;

  * $(\rm{2})$

$h\equiv 2\pmod{4}$ and $k\equiv 0\pmod{4}$;

  * $(\rm{3})$

$h\equiv 0\pmod{4}$ and $k\equiv 2\pmod{4}$;

  * $(\rm{4})$

$h,k\equiv 2\pmod{4}$ and $m,n$ both even.

###### Theorem 6.17

[35] There exists a ${}^{2}H(n;k)$ if and only if $n\geq k\geq 3$ with
$nk\not\equiv 1\pmod{4}$.

###### Theorem 6.18

[35] There exists a ${}^{2}H(m,n;n,m)$ if and only if $mn\not\equiv 1\pmod{4}$
and $m,n$ are not both equal to $2$.

###### Theorem 6.19

[35] Let $h,k$ be even positive integers. There exists a ${}^{2}H(m,n;h,k)$ if
and only if $mh=nk$, $2\leq h\leq n$, $2\leq k\leq m$ and $h,k$ are not both
equal to $2$. Moreover these arrays are integer when:

  * $(\rm{1})$

$h,k\neq 2$;

  * $(\rm{2})$

$h=2$ and either $n=2$ and $m=k\equiv 0\pmod{4}$ or $n,k\geq 3$;

  * $(\rm{3})$

$k=2$ and either $m=2$ and $n=h\equiv 0\pmod{4}$ or $m,h\geq 3$.

###### Proposition 6.20

[35] Let $n,k,\lambda$ be positive integers with $n\geq k\geq 3$ and $\lambda$
divisor of $k$. Then there exists a ${}^{\lambda}H_{\frac{k}{\lambda}}(n;k)$
if one of the following is satisfied:

  * $(\rm{1})$

$k=5$ and $n\equiv 3\pmod{4}$;

  * $(\rm{2})$

$k\neq 5$ odd and $n\equiv 0,3\pmod{4}$.

###### Proposition 6.21

[35] For any odd integer $n\geq 3$ and for any divisor $\lambda$ of $n$ there
exists a ${}^{\lambda}H_{\frac{n}{\lambda}}(n;3)$.

###### Proposition 6.22

[35] For any odd integer $n\geq 3$ and for any divisor $\lambda$ of $2n$ there
exists a ${}^{\lambda}H_{\frac{2n}{\lambda}}(n;3)$.

###### Proposition 6.23

[35] Let $m,n,h,k,t,\lambda$ be such that $mh=nk$, $4\leq h\leq n$, $4\leq
k\leq m$, $t$ is a divisor of $2nk$ and $\lambda$ is a divisor of $t$. A
${}^{\lambda}H_{\frac{t}{\lambda}}(m,n;h,k)$ exists in each of the following
cases:

  * $(\rm{1})$

$h\equiv k\equiv 0\pmod{4}$;

  * $(\rm{2})$

either $h\equiv 2\pmod{4}$ and $k\equiv 0\pmod{4}$ or $h\equiv 0\pmod{4}$ and
$k\equiv 2\pmod{4}$;

  * $(\rm{3})$

$h\equiv k\equiv 2\pmod{4}$ and $m$ and $n$ are even.

Some of previous results have been obtained thanks to one of the following two
theorems which allow one to obtain new arrays from existing ones.

###### Theorem 6.24

[35] If there exists an ${}^{\alpha}H_{t}(m,n;h,k)$ then, for any divisor
$\lambda$ of $t$, there exists a
${}^{\lambda\alpha}H_{\frac{t}{\lambda}}(m,n;h,k)$.

###### Theorem 6.25

[65] Let $m,n,h,k$ be such that $2\leq h\leq n$, $2\leq k\leq m$ and $mh=nk$.
Let $\lambda$ be a divisor of $2nk$ and let $t$ be a divisor of
$\frac{2nk}{\lambda}$. Set $d=\gcd(h,k)$. If there exists a (integer) diagonal
${}^{\lambda}H_{t}(\frac{nk}{d};d)$, then there exists a (integer)
${}^{\lambda}H_{t}(m,n;h,k)$.

Other existence results can be found using the following recursive
construction.

###### Proposition 6.26

[35] If there exists an (integer) $H_{t}(m,n;h,k)$ then there exists an
(integer)
${}^{\alpha_{1}\lambda_{1}}H_{t}(\lambda_{1}m,\lambda_{2}n;\alpha_{1}h,\alpha_{2}k)$
for all positive integers $\alpha_{1}\leq\lambda_{2}$,
$\alpha_{2}\leq\lambda_{1}$ such that
$\alpha_{1}\lambda_{1}=\alpha_{2}\lambda_{2}$, $\alpha_{1}h\leq\lambda_{2}n$
and $\alpha_{2}k\leq\lambda_{1}m$.

####  6.2.4 Connection between relative Heffter arrays and biembeddings and
graph decompositions

In [34, 35] the authors investigated the connection of these generalized
arrays with relative difference families, graph decompositions and
biembeddings. To illustrate the results we need some definitions. Given a
partially filled array $A$ by _skeleton_ of $A$, denoted by $skel(A)$, one
means the set of the filled positions of $A$, while by $\mathcal{E}(A)$ one
denotes the multiset of the elements of $skel(A)$. Analogously, by
$\mathcal{E}(R_{i})$ and $\mathcal{E}(C_{j})$ we mean the multisets of
elements of the $i$-th row and of the $j$-th column, respectively, of $A$. Now
we have to define a simple ordering of a _multiset_. We consider the elements
of $\mathcal{E}(A)$ indexed by the set $skel(A)$ (analogously for the rows and
the columns). For example let $A$ be the ${}^{4}H_{4}(4;2)$ constructed in
Example 6.6. Here $\mathcal{E}(A)=\\{-1,-1,1,1,-3,-3,3,3\\}$, we can view this
multiset as the set $\\{t_{b}\mid b\in
skel(A)\\}=\\{t_{(1,1)},t_{(1,2)},t_{(2,1)},t_{(2,2)},t_{(3,3)},t_{(3,4)},t_{(4,3)},t_{(4,4)}\\}$,
where $t_{(1,1)}=1$, $t_{(1,2)}=-1$, $t_{(2,1)}=-1$, $t_{(2,2)}=1$,
$t_{(3,3)}=3$, $t_{(3,4)}=-3$, $t_{(4,3)}=-3$ and $t_{(4,4)}=3$. Given a
finite multiset $T=[t_{b_{1}},\dots,t_{b_{k}}]$ whose (not necessarily
distinct) elements are indexed by a set $B=\\{b_{1},\dots,b_{k}\\}$ and a
cyclic permutation $\alpha$ of $B$ we say that the list
$\omega_{\alpha}=(t_{\alpha(b_{1})},t_{\alpha(b_{2})},\ldots,t_{\alpha(b_{k})})$
is the ordering of the elements of $T$ associated to $\alpha$. In case the
elements of $T$ belong to an abelian group $G$, we define
$s_{i}=\sum_{j=1}^{i}t_{\alpha(b_{j})}$, for any $i\in\\{1,\ldots,k\\}$, to be
the $i$-th partial sum of $\omega_{\alpha}$ and we set
$\mathcal{S}(\omega_{\alpha})=(s_{1},\ldots,s_{k})$. The ordering
$\omega_{\alpha}$ is said to be _simple_ if $s_{b}\neq s_{c}$ for all $1\leq
b<c\leq k$.

###### Proposition 6.27

If there exists a simple ${}^{\lambda}H_{t}(m,n;h,k)$, then there exist a
$(2mh+t,t,C_{h},\lambda)$-DF and a $(2nk+t,t,C_{k},\lambda)$-DF.

###### Proposition 6.28

If there exists a simple ${}^{\lambda}H_{t}(m,n;h,k)$, then there exist a
cyclic $C_{h}$-decomposition $\mathcal{D}$ and a cyclic $C_{k}$-decomposition
$\mathcal{D}^{\prime}$ both of ${}^{\lambda}K_{\frac{2nk+t}{t}\times t}$. If
$\lambda=1$ the decompositions $\mathcal{D}$ and $\mathcal{D}^{\prime}$ are
orthogonal.

Also for these generalized arrays to get biembeddings one has to find
compatible orderings. Note that in Section 3 we have defined compatible
orderings only when $\mathcal{E}(A)$ is a set. Given an $m\times n$ partially
filled array $A$, by $\alpha_{R_{i}}$ we will denote a cyclic permutation of
the nonempty cells of the $i$-th row $R_{i}$ of $A$ and by $\omega_{R_{i}}$
(omitting the dependence on $\alpha_{R_{i}}$) the associated ordering of
$\mathcal{E}(R_{i})$. Similarly, for the $j$-th column $C_{j}$ of $A$, we
define $\alpha_{C_{j}}$ and $\omega_{C_{j}}$.

If for any $i\in\\{1,\ldots,m\\}$ and for any $j\in\\{1,\ldots,n\\}$, the
orderings $\omega_{R_{i}}$ and $\omega_{C_{j}}$ are simple, we define the
permutation $\alpha_{r}$ of $skel(A)$ by
$\alpha_{R_{1}}\circ\ldots\circ\alpha_{R_{m}}$ and we say that the associated
action $\omega_{r}$ on $\mathcal{E}(A)$ is the simple ordering for the rows.
Similarly we define $\alpha_{c}=\alpha_{C_{1}}\circ\ldots\circ\alpha_{C_{n}}$
and the simple ordering $\omega_{c}$ for the columns. Now, given a
$\lambda$-fold relative Heffter array ${}^{\lambda}H_{t}(m,n;h,k)$, say $A$,
then the orderings $\omega_{r}$ (associated to the permutation $\alpha_{r}$)
and $\omega_{c}$ (associated to the permutation $\alpha_{c}$) are said to be
_compatible_ if $\alpha_{c}\circ\alpha_{r}$ is a cycle of length $|skel(A)|$.
In [35] the authors showed that the necessary conditions of Theorem 3.5 hold
more in general for a ${}^{\lambda}H_{t}(m,n;h,k)$.

###### Theorem 6.29

[35] Let $A$ be a $\lambda$-fold relative Heffter array
${}^{\lambda}H_{t}(m,n;h,k)$ that is simple with respect to the compatible
orderings $\omega_{r}$ and $\omega_{c}$. Then there exists a cellular
biembedding of a cyclic $C_{h}$-decomposition and a cyclic
$C_{k}$-decomposition both of ${}^{\lambda}K_{(\frac{2nk}{\lambda t}+1)\times
t}$ into an orientable surface.

###  6.3 Signed Magic Arrays

The notion of a signed magic array was introduced by Khodkar, Schulz and
Wagner [57] in 2017. These are intimately related to (relative) Heffter
arrays. We begin with the definition.

###### Definition 6.30

A _signed magic array_ $SMA(m,n;h,k)$ is an $m\times n$ array with entries
from $X$, where $X=\left\\{0,\pm 1,\pm 2,\ldots,\pm\frac{nk-1}{2}\right\\}$ if
$nk$ is odd and $X=\left\\{\pm 1,\pm 2,\ldots,\pm\frac{nk}{2}\right\\}$ if
$nk$ is even, such that:

  * $(\rm{a_{3}})$

each row contains $h$ filled cells and each column contains $k$ filled cells;

  * $(\rm{b_{3}})$

every integer from the set $X$ appears exactly once in the array;

  * $(\rm{c_{3}})$

the sum of each row and of each column is $0$ in $\mathbb{Z}$.

###### Remark 6.31

As was shown in [35], a signed magic array $SMA(m,n;h,k)$ with $nk$ even is an
integer ${}^{2}H(m,n;h,k)$, while in general the converse is not true. For
instance, the ${}^{2}H(6;4)$ of Example 6.7 is not a signed magic array. On
the other hand, in the particular case in which either $h=2$ or $k=2$ an
integer ${}^{2}H(m,n;h,k)$ is precisely an $SMA(m,n;h,k)$, see [35] for
details.

Existence results on signed magic arrays can be found in [53, 54, 55, 56, 57,
64, 65]. We note that nearly all the results and techniques used on signed
magic arrays are analogous to those for Heffter arrays. In particular, the
existence of an $SMA(m,n;h,k)$ has been determined in the square case and when
the array has no empty cells, as shown in the following two theorems.

###### Theorem 6.32

[57] There exists an $SMA(n,n;k,k)$ if and only if either $n=k=1$ or $3\leq
k\leq n$.

###### Theorem 6.33

[57] There exists an $SMA(m,n;n,m)$ if and only if one of the following cases
occurs:

  * $(1)$

$m=n=1$;

  * $(2)$

$m=2$ and $n\equiv 0,3\pmod{4}$;

  * $(3)$

$n=2$ and $m\equiv 0,3\pmod{4}$;

  * $(4)$

$m,n>2$.

Also the cases where each column contains 2 or 3 filled cells have been
solved.

###### Theorem 6.34

[53] There exists an $SMA(m,n;h,2)$ if and only if one of the following cases
occurs:

  * $(1)$

$m=2$ and $n=h\equiv 0,3\pmod{4}$;

  * $(2)$

$m,h>2$ and $mh=2n$.

###### Theorem 6.35

[56] There exists an $SMA(m,n;h,3)$ if and only if $3\leq m$, $h\leq n$ and
$mh=3n$.

Necessary and sufficient conditions have been established also in the case
when $h$ and $k$ are both even.

###### Theorem 6.36

[64] Let $h,k$ be two even integers with $h,k\geq 4$. There exists an
$SMA(m,n;h,k)$ if and only if $4\leq h\leq n$, $4\leq k\leq m$ and $mh=nk$.

Recent partial results for the general case can be found in [65]. We summarize
them in the following two theorems.

###### Theorem 6.37

[65] Let $m,n,h,k$ be four integers such that $3\leq h\leq n$, $3\leq k\leq m$
and $mh=nk$. There exists an $SMA(m,n;h,k)$ whenever $\gcd(h,k)\geq 2$.

###### Theorem 6.38

[65] Let $m,n,h,k$ be four integers such that $3\leq h\leq n$, $3\leq k\leq m$
and $mh=nk$. If $h\equiv 0\pmod{4}$, then there exists an $SMA(m,n;h,k)$.

###  6.4 Non-zero sum Heffter arrays

While studying the conjecture of Alspach (Conjecture 4.3), the authors of [30]
proposed the following variant of a Heffter array which was termed a non-zero
sum Heffter array. The formal definition follows.

###### Definition 6.39

Let $v=\frac{2nk}{\lambda}+t$ be a positive integer, where $t$ divides
$\frac{2nk}{\lambda}$, and let $J$ be the subgroup of $\mathbb{Z}_{v}$ of
order $t$. A $\lambda$-_fold non-zero sum Heffter array $A$ over
$\mathbb{Z}_{v}$ relative to $J$_, denoted by ${}^{\lambda}NH_{t}(m,n;h,k)$,
is an $m\times n$ array with elements in $\mathbb{Z}_{v}$ such that:

  * $(\rm{a_{4}})$

each row contains $h$ filled cells and each column contains $k$ filled cells;

  * $(\rm{b_{4}})$

the multiset $\\{\pm x\mid x\in A\\}$ contains each element of
$\mathbb{Z}_{v}\setminus J$ exactly $\lambda$ times;

  * $(\rm{c_{4}})$

the sum of the elements in every row and column is different from $0$ in
$\mathbb{Z}_{v}$.

If the array is square, then $h=k\geq 1$, while if $m\neq n$ then at least one
of $h$ and $k$ has to be greater than $1$. A square non-zero sum Heffter array
is denoted by ${}^{\lambda}NH_{t}(n;k)$. If $\lambda=1$ or $t=1$, then it is
omitted.

###### Example 6.40

An $NH(7;5)$ over $\mathbb{Z}_{71}$:

$\begin{array}[]{|r|r|r|r|r|r|r|}\hline\cr 10&&16&-1&-2&-3&\\\
\hline\cr&4&&-6&-7&-5&22\\\ \hline\cr-30&29&9&-8&&&18\\\
\hline\cr-11&&-12&-28&-31&26&\\\ \hline\cr&-14&-15&-13&&-17&25\\\ \hline\cr
27&-34&20&&-19&&-32\\\ \hline\cr 24&23&&&21&-35&33\\\ \hline\cr\end{array}$

In [30] the authors focused on the case in which $\lambda=1$ and they proved
that the trivial necessary conditions are also sufficient in the very general
case in which the array is rectangular, empty cells are allowed and for every
arbitrary subgroup $J$. We would like to note, however, that in the square
case the proof of existence is quite easy when there exists an $H_{t}(n;k)$.
Begin with the $H_{t}(n;k)$ and find a transversal of filled cells $T$ (this
must exist). Then change the sign of the element in each of the cells of $T$
and note that now no row or column can add to $0$. Hence we have constructed
an $NH_{t}(n;k)$. The general case is more involved and we emphasize that the
proof of the general result is not constructive.

###### Theorem 6.41

[30] There exists an $NH_{t}(m,n;h,k)$ if and only if $mh=nk$, $m\geq k\geq
1$, $n\geq h\geq 1$ and $t$ divides $2nk$.

These arrays have been introduced since, as with classical ones, they are
related to open problems on partial sums, difference families, graph
decompositions and biembeddings. The interested reader is referred to [30]
where all the following results can be found.

By reasoning similar to what was done in Section 5, it is easy to see that if
a row (column, respectively) of a $NH_{t}(m,n;h,k)$ admits an ordering whose
partial sums are pairwise distinct and non-zero then one can construct a path
of length $h$ ($k$, respectively). For example, starting from the first row of
the $NH(7;5)$ of Example 6.40 and considering its natural ordering one obtains
the path $\Gamma=[0,10,26,25,23,20]$ of length $5$, clearly $\Delta\Gamma=\pm
R_{1}$. A non-zero sum Heffter array is said to be simple if every row and
every column admits an ordering such that the partial sums are pairwise
distinct and non-zero. Hence, since every row and column of a non-zero sum
Heffter array can be viewed as a set $T$ considered in Conjecture 4.3, we have
that if Conjecture 4.3 were true, then every non-zero sum Heffter array would
be simple. Investigating arrays having this additional property is important
in view of the following results.

###### Proposition 6.42

[30] If there exists a simple ${}^{\lambda}NH_{t}(m,n;h,k)$, then there exist
a $(2mh+t,t,P_{h},\lambda)$-DF and a $(2nk+t,t,P_{k},\lambda)$-DF.

###### Proposition 6.43

[30] If there exists a simple ${}^{\lambda}NH_{t}(m,n;h,k)$, then there exist
a cyclic $P_{h}$-decomposition $\mathcal{D}$ and a cyclic
$P_{k}$-decomposition $\mathcal{D}^{\prime}$ both of
${}^{\lambda}K_{\frac{2nk+t}{t}\times t}$. If $\lambda=1$ the decompositions
$\mathcal{D}$ and $\mathcal{D}^{\prime}$ are orthogonal.

In view of previous propositions, in [30, 59] the authors focused on globally
simple non-zero sum Heffter arrays, obtaining the following existence results.
All the proofs are constructive.

###### Theorem 6.44

[30] For every $n\geq k\geq 1$ there exists a globally simple cyclically
$k$-diagonal $NH(n;k)$.

###### Theorem 6.45

[30] For every $m,n\geq 1$ there exists a globally simple $NH(m,n;n,m)$.

###### Theorem 6.46

[59] For every odd integer $n\geq 1$ and for every divisor $t$ of $n$, there
exists a globally simple $NH_{t}(n;n)$.

###### Proposition 6.47

[59] For every odd integer $n\geq 1$ and for every
$t\in\\{2,2n,n^{2},2n^{2}\\}$, there exists a globally simple $NH_{t}(n;n)$.

By Theorems 6.44 and 6.46 and Proposition 6.47 one gets the following result
(the case $n=2$ is trivial).

###### Corollary 6.48

[59] Let $n$ be a prime. There exists a globally simple $NH_{t}(n;n)$ for
every admissible $t$.

The case of $\lambda$-fold non-zero sum Heffter arrays with $\lambda>1$ has
been investigated in [28]. Here the authors consider a generic finite group
$G$ and prove that there exists a non-zero sum $\lambda$-fold Heffter array
over $G$ relative to $J$ whenever the trivial necessary conditions are
satisfied and $|G|\geq 41$. Moreover, this value can be decreased to $29$ when
the array does not contain empty cells.

We conclude this subsection with the theorem that gives the connection to
biembeddings.

###### Theorem 6.49

[30] Let $A$ be a non-zero sum Heffter array $NH(m,n;h,k)$ with compatible
orderings $\omega_{r}$ and $\omega_{c}$. Then there exists a cellular
biembedding of two circuit decompositions of $K_{2nk+1}$ into an orientable
surface, such that the faces are multiples of $h$ and $k$ strictly larger than
$h$ and $k$, respectively.

Details on the faces of the biembeddings can be given considering the order of
the row and column sums in the group, as done in [25, 59].

###  6.5 Heffter configurations

Heffter arrays are generalized to Heffter configurations in [18]. We recall
that a $(v,b,k,r)$-configuration is a pair $(V,{\cal B})$ where $V$ is a set
of $v$ points and $\cal B$ is a $b$-set of $k$-subsets (blocks) of $V$ such
that any two distinct points are contained together in at most one block and
any point occurs in exactly $r$ blocks. Such a configuration is resolvable if
there exists a partition of $\cal B$ into $r$ parallel classes each of which
is a partition of $V$, see [19, 43]. Also, it is $\mathbb{Z}_{v}$-additive if
its point set is a subset of $\mathbb{Z}_{v}$ and each block sums to zero.

###### Definition 6.50

A $(nk,nr,k,r)$ _Heffter configuration_ (briefly HC) is a
$\mathbb{Z}_{2nk+1}$-additive resolvable $(nk,nr,k,r)$-configuration where the
point set is a half-set of $\mathbb{Z}_{2nk+1}$. It is proper if $r\geq 3$ and
it is simple if each block admits a simple ordering.

The above terminology is justified by the fact that a $(nk,nr,k,r)$-HC is
equivalent to $r$ mutually sub-orthogonal Heffter systems on the same half-set
of $\mathbb{Z}_{2nk+1}$. Indeed each parallel class is nothing but a Heffter
system D$(nk,k)$. Hence, a Heffter system D$(nk,k)$ is a $(nk,n,k,1)$-HC and
an Heffter array $H(n;k)$, say $A$, is a $(nk,2n,k,2)$-HC whose two parallel
classes are the rows of $A$ and the columns of $A$.

The construction of proper HCs appears to be very hard. On the other hand some
infinite series have been obtained recursively by means of small examples such
as the following.

###### Example 6.51

The integer shiftable $H(5;4)$ below gives a $(20,15,4,3)$-HC whose three
parallel classes are: the rows; the columns; and the diagonals
$D_{1},\ldots,D_{5}$.

$\begin{array}[]{|r|r|r|r|r|}\hline\cr-1&2&17&-18&\\\ \hline\cr
6&10&-13&&-3\\\ \hline\cr 9&-5&&-8&4\\\ \hline\cr-14&&-16&11&19\\\
\hline\cr&-7&12&15&-20\\\ \hline\cr\end{array}$

Generalizing Proposition 5.9, we can state the following.

###### Theorem 6.52

A simple $(nk,nr,k,r)$-HC gives rise to $r$ mutually orthogonal cyclic
$C_{k}$-decompositions of $K_{2nk+1}$.

###  6.6 Further generalizations

Another natural generalization is to work in an arbitrary additive group $G$
and not necessarily in the cyclic group. Some work in this direction has been
done in [37] where the authors proposed the very general concept of an
_Archdeacon array_.

###### Definition 6.53

[37] An _Archdeacon array_ over an abelian group $(G,+)$ is an $m\times n$
array $A$ with elements in $G$, such that:

  * $(\rm{b_{5}})$

the multiset $\\{\pm x\mid x\in A\\}$ contains each element of
$G\setminus\\{0\\}$ at most once;

  * $(\rm{c_{5}})$

the elements in every row and column sum to $0$ in $G$.

###### Example 6.54

An Archdeacon array on $\mathbb{Z}_{41}\oplus\mathbb{Z}_{d}$, for $d\geq 3$:

$\begin{array}[]{|r|r|r|r|r|}\hline\cr(-1,1)&(0,-1)&(-14,0)&(9,0)&(6,0)\\\
\hline\cr(2,-1)&(-7,1)&&(-5,0)&(10,0)\\\
\hline\cr(17,0)&(12,0)&(-16,0)&&(-13,0)\\\
\hline\cr(-18,0)&(15,0)&(11,0)&(-8,0)&\\\
\hline\cr&(-20,0)&(19,0)&(4,0)&(-3,0)\\\ \hline\cr\end{array}$

Clearly, Heffter arrays are a particular kind of Archdeacon arrays. Examples,
existence results and applications of these arrays can be found in Section $5$
of [37].

Another general concept termed a quasi-Heffter array has recently been
proposed in [25]. In that paper the reader can find all the details on the
applications of these arrays to biembeddings and their full automorphism
group.

###### Definition 6.55

[25] Let $v=2nk+t$ be a positive integer, where $t$ divides $2nk$, and let $J$
be the subgroup of $\mathbb{Z}_{v}$ of order $t$. A _quasi-Heffter array_ $A$
_over $\mathbb{Z}_{v}$ relative to $J$_, denoted by $QH_{t}(m,n;h,k)$, is an
$m\times n$ array with elements in $\mathbb{Z}_{v}$ such that:

  * $(\rm{a_{6}})$

each row contains $h$ filled cells and each column contains $k$ filled cells;

  * $(\rm{b_{6}})$

the multiset $\\{\pm x\mid x\in A\\}$ contains each element of
$\mathbb{Z}_{v}\setminus J$ exactly once.

So in this case the definition does not require any property on the row and
column sums. In general, given $S\subseteq\mathbb{Z}_{v}$ one may require that
the sums of the rows and of the columns belong to $S$. Clearly, if $S=\\{0\\}$
we find the classical concept of a Heffter array and if
$S=\mathbb{Z}_{v}\setminus\\{0\\}$ we get the notion of a non-zero sum Heffter
array. For other choices of $S$ we have further variants of the classical
concept having again applications to difference families, graph decompositions
and biembeddings.

##  7 Conclusions and Open Problems

Heffter arrays are a relatively new type of combinatorial design with original
paper on this topic published in 2015. Since that time there have been a
number of papers published on this subject as well as on topics that are
related to Heffter arrays. Here we have surveyed all the major results of
these papers. We believe that there are many more interesting questions
relating to Heffter arrays and related subjects and we fully expect
researchers to make good progress on these in the years to come. We conclude
this survey with a short list of open problems.

  * •

Complete the construction of classical Heffter arrays in the rectangular case
with empty cells, for known results see Theorems 1.17 and 1.18.

  * •

Construct new infinite classes of globally simple Heffter arrays, for known
results see Theorems 2.5, 2.6 and 2.7.

  * •

Determine new solutions to the Crazy Knight’s Tour Problem, see Section 3.

  * •

Determine new results concerning Conjectures 4.3, 4.4 and 4.5.

  * •

Establish more restrictive necessary conditions for integer
${}^{\lambda}H_{t}(n;k)$, in order to improve Proposition 6.10.

###  7.1 Acknowledgements

The authors would like to thank the Fields Institute for supporting the
Stinson66 Conference and for financial support to the authors to attend the
conference.

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