Weak Heffter Arrays and biembedding graphs on non-orientable surfaces

In 2015, Archdeacon proposed the notion of Heffter arrays in view of its connection to several other combinatorial objects. In the same paper he also presented the following variant. A weak Heffter array $\mathrm{W}\mathrm{H}(m,n;h,k)$ is an $m \times n$ matrix $A$ such that: each row contains $h$ filled cells and each column contains $k$ filled cells; 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; the elements in every row and column (with the corresponding sign) sum to $0$ in $\mathbb{Z}_{2nk+1}$. Also the ``weak concept'', as the classical one, is related to several other topics, such as difference families, cycle systems and biembeddings. Many papers on Heffter arrays have been published, while no one on weak Heffter arrays has been written. This is the first one and here we explore necessary conditions, existence and non-existence results, and connections to biembeddings into non-orientable surfaces.


Introduction
In 2015, Archdeacon [1] introduced a class of partially filled (p.f. for short) arrays as an interesting link between combinatorial designs and topological graph theory.Since then, 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.Definition 1.1.[1] A Heffter array H(m, n; h, k) is an m × n matrix with entries from Z 2nk+1 such that: (a) each row contains h filled cells and each column contains k filled cells; (b) for every x ∈ Z 2nk+1 \ {0}, either x or −x appears in the array; (c) the elements in every row and column sum to 0 in Z 2nk+1 .
To date several variants and generalizations of this concept have been introduced (see [17]), so we will refer to these arrays as classical.
In [1], Archdeacon also proposed the following variant of the previous concept.
Definition 1.2.A weak Heffter array WH(m, n; h, k) is an m × n matrix A such that: (a 1 ) each row contains h filled cells and each column contains k filled cells; (b 1 ) for every x ∈ Z 2nk+1 \ {0}, there is exactly one cell of A whose element is one of the following: x, −x, ±x, ∓x, where the upper sign on ± or ∓ is the row sign and the lower sign is the column sign; (c 1 ) the elements in every row and column (with the corresponding sign) sum to 0 in Z 2nk+1 .
If m = n, then h = k and we use the notation H(n; k) instead of H(n, n; k, k).Furthermore, a rectangular array with no empty cells H(m, n; n, m) is denoted by H(m, n).Analogous notation holds for a weak Heffter array.
A (weak) Heffter array is said to be integer if condition (c) of Definition 1.1 (condition (c 1 ) of Definition 1.2) is strengthened so that the elements in every row and every column, seen as integers in ±{1, . . ., nk}, sum to zero in Z.
Example 1.3.The following is a weak Heffter array WH (3,4) over Z 25 (see Figure 6 of [1]): The existence of classical Heffter arrays has been largely investigated.The most important results are the following.
Theorem 1.5.[2] There exists an H(m, n) if and only if m, n ≥ 3.
Recent partial results for integer rectangular Heffter arrays with empty cells can be found in [16].
On the other hand, no result has been obtained about weak Heffter arrays, actually, this is the first paper studying this class of arrays.As shown into details in Section 2, one can get a weak Heffter array starting from a classical one simply by placing different row and column signs in a suitable set of cells, see for instance, Example 2.2.Clearly, not all weak Heffter arrays can be constructed starting from a classical one.Indeed the one proposed by Archdeacon, see Example 1.3, that as far as we know is the only weak Heffter array considered, cannot be obtained from a classical H (3,4) by placing different row and column signs in a suitable set of cells.This induces us to introduce the following definition.Definition 1.6.A weak WH(m, n; h, k) is said to be strictly weak if it cannot be obtained from a classical H(m, n; h, k) by placing different row and column signs in a suitable set of cells.
In other words, a weak Heffter array A = (a ij ) is strictly weak if there is no Heffter array B = (b ij ) with the same parameters such that |a ij | = |b ij |, for any i, j.Hence it is easy to see that the WH (3,4) of Example 1.3 is strictly weak.The terminology strictly weak may induce the reader to believe it is easier to construct these arrays than the classical ones.However, as shown in Section 3, this is not the case.Notably, for some values of n and k, an H(n; k) exists while a strictly weak WH(n; k) does not exist.The motivation for which Archdeacon introduced this variant and for which we believe it is worth studying is that, as shown in Section 6.1 of [17], also these arrays give rise to orthogonal cyclic cycle systems and, as hinted in [1] and explained in details in Section 5, starting from these arrays the existence of embeddings follows.
In Section 2 we investigate the connection between classical and (not strictly) weak Heffter arrays, showing how some existence results on the classical case apply to this context.In Sections 3 and 4 we consider and adjust to the weak context a more general concept, introduced in [7], which admits classical Heffter arrays as a particular case: the relative Heffter arrays.Definition 1.7.Let v = 2nk + t be a positive integer, where t divides 2nk, and let J be the subgroup of Z v of order t.A Heffter array A over Z v relative to J, denoted by H t (m, n; h, k), is an m × n array with elements in Z v \ J such that: (a 2 ) each row contains h filled cells and each column contains k filled cells; (b 2 ) for every x ∈ Z v \ J, either x or −x appears in the array; (c 2 ) the elements in every row and column sum to 0 in Z v .
Analogously, one can define a weak relative Heffter array WH t (m, n; h, k), it is sufficient to replace condition (b 2 ) of Definition 1.7 with the following one: (b 3 ) for every x ∈ Z v \ J, there is exactly one cell of A whose element is one of the following: x, −x, ±x, ∓x, where the upper sign on ± or ∓ is the row sign and the lower sign is the column sign.Clearly, if t = 1 we find again the original concepts introduced in [1].For the square case we use the notation H t (n; k) and WH t (n; k).
In details, in Section 3 we determine some necessary conditions for the existence of a weak (relative) Heffter array and then we present some non-existence results.Furthermore, in Propositions 3.9 and 3.10 we show that for some values of t, n and k there exists an H t (n; k) while no strictly weak WH t (n; k) exists.In Section 4 we construct an infinite class of strictly weak relative Heffter arrays for some values of the parameters for which the existence of the corresponding relative Heffter array was left open in [7].In Sections 5 and 6, we investigate the relation between weak Heffter arrays and embeddings.Indeed while the connection between classical Heffter arrays and embeddings into orientable surfaces has been considered by several papers (see, for instance, [5,8,9,10]), the one between weak Heffter arrays and embeddings has been only marginally investigated in [1].Here, in Section 5 we provide a formal definition of Archdeacon embedding into non-necessarily orientable surfaces.Then, in Section 6, we present an infinite family of non-orientable embeddings of Archdeacon type.In the last section we conclude with some remarks and we present an open problem.

Connection between weak and classical Heffter arrays
As suggested by the terminology, a weak Heffter array is an Heffter array in which a property is relaxed.So we believe it is quite natural to ask if it is possible to construct a weak Heffter array starting from a classical one.Clearly, given a Heffter array A if we replace all the elements a of A with ±a we trivially get a weak Heffter array.It is also easy to see that given a set R of rows of A if we replace each element a of R with ∓a we get a weak Heffter array.Obviously a similar reasoning can be done on the columns.
In what follows, given an array A, we respectively denote by R i and C j the i-th row and the j-th column of A. Then, by support of A we mean the set of the absolute values of the elements contained in A, that is supp(A) = {|a| : a ∈ A}.We point out that in this section, with a little abuse of notation, we identify a row (column) of a (weak) H(m, n; h, k) with the h-subset (k-subset) of Z 2nk+1 whose elements are those of the given row (column).
The following result shows a way to get a weak Heffter array, starting from a classical one, having few cells with different row and column signs.Proposition 2.1.If there exists an H(m, n; h, k) with a row or a column containing a proper subset whose sum is zero (modulo 2nk + 1), then there exists a WH(m, n; h, k) whose number of cells containing an element with different row and column signs is at most equal to max h 2 , k 2 .
Proof.Let A be an array as in the statement.It is not restrictive to reason on a row.So let S be a proper subset of a row R of A summing up to zero.Since every row sums to zero also the elements of R \ S sum to zero, hence we can suppose that |S| ≤ h 2 .Now we replace each element in S by ∓s, and we leave unchanged all the other elements in A. Call B the new array.Clearly, supp(A) = supp(B) and all the rows of B different from R are nothing but the rows of A, hence their sums are zero.About R, the sum is still zero since we have changed the signs of the elements of a subset of R whose sum is zero.Finally, note that the column signs are not changed.So the columns of B sum to 0. We believe that it is natural to ask if some of the classical Heffter arrays have the property of Proposition 2.1.First of all note that since a row (column) of an H(m, n; h, k) does not contain 0 nor 2-subsets of the form {x, −x}, if the property is satisfied by a row (column) then h ≥ 6 (k ≥ 6).We recall that in all the constructions of square integer Heffter arrays, see [3,11], the authors obtain an H(n; k + 4) starting from an H(n; k) by adding to each row and each column four elements having sum zero.So the condition required by Proposition 2.1 is trivially satisfied.Hence we get the following.
Corollary 2.3.For every n > k ≥ 3, there exists an integer WH(n; k) with exactly ℓ cells containing an element with different row and column signs, where Clearly all the arrays constructed with one of the techniques illustrated in this section are, by definition, not strictly weak.

Necessary conditions for weak Heffter arrays and non-existence results
In this section we present some non-existence results on (strictly) weak Heffter arrays.It is important to underline that the existence of a strictly weak Heffter array is not always granted.In fact, it seems that, despite having an higher degree of freedom in the choice of the signs of the elements in the array, strictly weak Heffter arrays are as difficult to find as classical ones.
Furthermore, it may happen that there exists a Heffter array but not a strictly weak Heffter array with the same parameters, as shown in the following remark.
Remark 3.1.By Theorem 1.4 an H(n; k) exists for any n ≥ k ≥ 3. On the other hand we have checked, with the aid of a computer, that no strictly WH(n; 3) exists when n = 3, 4.
Remark 3.2.We point out that if there is no weak Heffter array this means there is also no classical or strictly weak Heffter array with the same parameters.While if there is no strictly weak Heffter array, this gives no information about the existence of a classical or weak Heffter array with the same parameters.
We start with a preliminary consideration.Proposition 3.3.A weak Heffter array WH(m, n; h, k) has at least 3 cells containing distinct row and column signs.
Proof.In this proof given an array A by a R ij (respectively, a C ij ) we denote the element of A in the position (i, j) with its row (respectively, column) sign.Also, by A R (respectively, A C ) we mean the sum of all the elements in A with their row (respectively, column) sign.Set From this, it follows that |I| = 1 since 0 ∈ A and Z 2nk+1 does not contain the involution.Also |I| = 2 since A does not contain opposite elements and, again, the involution does not exist.The thesis follows.
In the remain of this section we propose some non-existence results for weak Heffter arrays and we compare the existence of classical and strictly weak Heffter arrays for the same given class of parameters.
Remark 3.4.A weak relative Heffter array cannot have exactly one cell with different row and column signs since it does not contain 0 and the involution (if it exists).On the other hand, a weak relative Heffter array may have exactly two cells with different row and column signs, as in the following example, that is a WH 16 (4; 4) (rielaboration of Example 1.4 of [7]): To present our non-existence results we have to recall that the rows and the columns of a (weak) Heffter array give two Heffter systems.Here we present a generalized definition in which we consider a non-trivial subgroup of a cyclic group.Definition 3.5.A Heffter system D t (v; k) is a set of zero-sum k-subsets (called blocks) of Z v such that for every x ∈ Z v \ J, where J is the subgroup of Z v of order t, exactly once of each pair {x, −x} is contained in exactly one block.
If t = 1 we find the classical concept of a Heffter system, see [13].It is clear that given a subgroup J of Z v , the existence of a Heffter system on Z v \ J is a necessary condition for the existence of a (weak) Heffter array on Z v relative to J.This fact leads us to generalise the statement of Lemma 3.3 of [7], and to obtain a non-existence result of weak Heffter arrays for an infinite family of parameters.
Proof.Assume that there exists such a Heffter system, denoted by D. It is easy to see that its support is 1, 2, 3, . . ., ⌊ 9n 2 ⌋ \ 3, 6, 9, . . ., 3⌊ 3n 2 ⌋ .For every triple {a 1 , a 2 , a 3 } ∈ D, the sum of its elements must be zero in Z 9n , and since none of them can be equal to zero modulo 3, they must belong to the same residue class modulo 3.
Moreover, as for every element a in D its opposite −a is not contained in D, we may assume that every triple of D contains elements all congruent to 1 modulo 3.In fact, if any given triple However, the elements in Z 9n \ J equal to 1 modulo 3 are: It is then evident that, in any case, these elements cannot sum to zero in Z 9n , giving a contradiction.
Proof.The result immediately follows from previous lemma.
We point out that there are examples where there exists exactly one Heffter system with some given parameters.Hence, also in these cases, it is not possible to construct a Heffter array, neither classical nor weak.Finally, also the existence of at least two Heffter systems is not a sufficient condition for the existence of a Heffter array, see Proposition 3.10.Now, as a consequence of Remark 3.1, we believe that it is interesting to study the existence problem of a WH t (n; 3) when n = 3, 4 for every admissible t.Firstly, we present the following result whose proof relies on the fact that, if we are working in Z or in Z v with v even, the array must contain an even number of odd numbers.The proof is omitted since it is similar to that of Proposition 3.1 of [7].
Proposition 3.8.Suppose that there exists an integer WH t (n; k), or a non-integer WH t (n; k) with t even.
( (1) The existence of an H t (3; 3) is known for t = 1, see Theorem 1.4, and t = 3, see Theorem 1.5 of [7].The following is an H 6 (3; 3): (2) For t = 1 a strictly WH t (3; 3) does not exist by Remark 3.1.For t = 3 there are only the following four Heffter systems D 3 (21; 3): We have checked, with the aid of a computer, that using them it is not possible to construct a strictly weak Heffter array.Finally, the following is a strictly WH 6 (3; 3): (2) For t = 1 a strictly WH t (4; 3) does not exist by Remark 3.1.The following are a strictly WH 2 (4; 3) and a strictly WH 4 (4; 3) : For t = 3, 6 there are respectively 9 and 10 Heffter systems D t (24 + t; 3), and we have checked using a computer that it is not possible to construct a strictly WH t (4; 3) starting from them.

An infinite class of strictly weak Heffter arrays
In this section we construct an infinite family of strictly weak Heffter arrays for a class of parameters for which, at the moment, the existence of a Heffter array is unknown.
Note that for k = 5 the existence problem of integer relative Heffter arrays H 5 (n; 5) has been proved only for n ≡ 3 (mod 4), leaving the case n ≡ 0 (mod 4) open.For this class in [7] there are only two examples for n = 8 and n = 16.Here we focus on this open case and we prove that there exists a strictly weak integer WH 5 (n; 5) for every n ≡ 0 (mod 4), with n ≥ 12.
The construction we are going to present is based on filling in the cells of a set of diagonals.We recall that, given a square array of order n, for i = 1, . . ., n, the i-th diagonal is so defined D i = {(i, 1), (i + 1, 2), . . ., (i − 1, n)}.All the arithmetic on row and column indices is performed modulo n, where the set of reduced residues is {1, 2, . . ., n}.The diagonals D i , D i+1 , . . ., D i+k−1 are k consecutive diagonals.Given n ≥ k ≥ 1, a partially filled array A of order n is k-diagonal if its non empty cells are exactly those of k diagonals.Furthermore, if these diagonals are consecutive A is said to be cyclically k-diagonal.
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.In [11] the authors introduced the following smart notation to describe a diagonal array; we point out that this notation has been used in several subsequent papers including [7,9].This procedure for filling a sequence of cells on a diagonal is termed diag and it has six parameters, as follows.
Let A be an n × n p.f. array; then, the procedure diag(r, c, s, ∆ 1 , ∆ 2 , ℓ) fills the entries The parameters used in the diag procedure have the following meaning: • r denotes the starting row, • c denotes the starting column, • s denotes the entry A[r, c], • ∆ 1 denotes the increasing value of the row and column at each step, • ∆ 2 denotes how much the entry is changed at each step, • ℓ is the length of the chain.
In the following given two positive integers a, b with a ≤ b, by [a, b] we denote the set {a, a + 1, . . ., b}.We are now ready to present the main existence result of this section.We remark that in Example 4.3 we follow step by step the construction illustrated in the proof of the following theorem.Hopefully it can help the reader to understand the idea developed in the proof.Theorem 4.2.There exists a strictly weak integer WH 5 (n; 5) for every n ≡ 0 (mod 4) with n ≥ 12.
Proof.We begin by considering the integer cyclically 3-diagonal H 3 (n; 3) with n ≡ 0 (mod 4) constructed in Proposition 5.3 of [7], so let A be the n × n array built using the following procedures labeled A to J:

4
; We also fill the following cells of A in an ad hoc manner: Note that the filled cells of A are exactly those of the diagonals D 1 , D 2 and D n .Also, it is easy to see that: Consider now the matrix B obtained from A by adding 4n + 2 to the positive elements of D 1 and −(4n + 2) to the negative elements of D 1 .Since we have only changed the elements in the main diagonal of A, it follows that the total sum of the i-th row of B is equal to the total sum of the i-th column of B, for every i ∈ [1, n].In particular, their sum is 4n + 2 if i ∈ 2, n 2 + 1 , and , while the support of D 2 and D n is unchanged.Hence Consider the following arrays: ± n 2 + 2 and for any i ∈ 3, 5, 7, . . ., n 2 − 1 ∪ n 2 + 3, n 2 + 5, . . ., n − 1 , define M i to be: Set I = {1, 2} ∪ 3, 5, 7, . . ., n 2 − 1 ∪ n 2 + 3, n 2 + 5, . . ., n − 1 .It can then be seen that ).Note that, in such a way, we added two filled cells in each row and each column.Hence every row and every column of C has exactly 5 filled cells.Also note that Now, notice that the ordered list of the sums of the rows of this new array C, that is the same of the sum of its columns, is: ).
To conclude the construction, we need to exchange some elements belonging to the main diagonal of C. In particular, for every i ∈ { Notice that the ordered list of the total sum of the rows and that of the columns of C is: (0, 0, 0, 0, 0, 0, +1, +1, +1, −1, +1, −1).Hence, we need to apply the final readjustment of some elements of the main diagonal of C to gain the zero sum on every row and column ( It can now be seen that the obtained array is a strictly WH 5 (12; 5).

The Archdeacon Embedding
This section focuses mainly on the connection between partially filled arrays and embeddings.To explain this link, we first recall some basic definitions, see [14,15].Definition 5.1.Given a graph Γ and a surface Σ, an embedding of Γ in Σ is a continuous injective mapping ψ : Γ → Σ, where Γ is viewed with the usual topology as 1-dimensional simplicial complex.
The connected components of Σ \ ψ(Γ) are said ψ-faces.Also, with abuse of notation, we say that a circuit F of Γ is a face (induced by the embedding ψ) if ψ(F ) is the boundary of a ψ-face.Then, if each ψ-face is homeomorphic to an open disc, the embedding ψ is called cellular.
If properties (a) and (b) hold, the map ρ is said to be a rotation of Γ.
Then, as reported in [12], a combinatorial embedding Π = (Γ, ǫ, ρ) is equivalent to a cellular embedding ψ of Γ into a surface Σ.Now we revisit the definition of the Archdeacon embedding in the case of weak Heffter arrays.We first introduce some notation.Given a partially filled array A, we denote by E(A), E(R i ), E(C j ) the list of the elements of the filled cells of A, of the i-th row and of the j-th column, respectively.By ω Ri and ω Cj we mean an ordering of, respectively, E(R i ) and E(C j ).We then define by Recalling that the rows (resp.columns) are obtained by considering the upper (resp.lower) signs, we have that Here if we consider the natural orderings, from left to right for the rows and from top to bottom for the columns, we have that Let A be a WH(m, n; h, k), we denote by Θ(A) the m × n array obtained by deleting from A the elements of type ±x or ∓x.Then we denote by Ω(A) the complement of Θ(A) in A. We note that Therefore, we can define the map λ : Example 5.4.We consider again the array A of Example 1.3.Here we have that: (3,4), Here we have that λ(x) = 1 whenever x ∈ ±{1, −7, −6, 12, 2, −4, −3, 5} and −1 otherwise.
Definition 5.5.Let A be a WH(m, n; h, k) and let us consider the ordering ω r for the rows and ω c for the columns.Note that we can assume, without loss of generality, that the cell (1, 1) is filled.We denote by a 1 the element contained in the cell (1, 1), considered with its row sign so that ω r (a 1 ) and ω −1 r (a 1 ) are welldefined.Then, we define recursively where µ 1 = λ(a 1 ) and From now on we will use the notation (b i ) to indicate the sequence of elements (b 1 , b 2 , • • • ).In this discussion we prefer to avoid introducing the concept of a current graph, but, following [1], the sequence (a i ) obtained from Equation (5.1) defines the face in the embedding of the current graph, see [12], constructed (with the face-trace algorithm) starting from the edge associated with a 1 .Here, using the notations of [1], the sequence (µ i ) defined in Equation (5.2) keeps track of the local sense of orientation (that is either anticlockwise or clockwise).Proposition 5.6.Given a WH(m, n; h, k), the following facts are equivalent: (1) The sequence ((a i , µ i )) has period 2nk; (2) The first 2nk elements of (a i µ i ) are all distinct; (3) The sequence (a i µ i ) has period 2nk; (4) The sequence ((a 2i+1 , µ 2i+1 )) has period nk.
Proof.Let A be a WH(m, n; h, k).Firstly, note that from Equation (5.1), it follows that: (1) ⇒ (2).Let us suppose, by contradiction, that (1) holds but (2) does not.This means that there exist i and j, with 2nk ≥ j > i ≥ 1 (which implies j − i < 2nk), such that a i µ i = a j µ j .Also, let i and j be indexes with this property with minimum difference.Due to the definition of the sequences (a i ) and (µ i ), if a i = a j , µ i = µ j and i ≡ j (mod 2), then T |(j − i) where T = 2nk is the period of the sequence ((a i , µ i )).Moreover, because of the definition of (a i ), if a i = −a j we must have that i ≡ j (mod 2).In both cases, if a i µ i = a j µ j and j − i < 2nk, we must have that i ≡ j (mod 2).Furthermore, since in each row and each column we have more than one element, j − i > 1.
Here we have four possible cases: a) (a i , µ i ) = (a j , µ j ), i is odd and j is even; b) (a i , µ i ) = (a j , µ j ), i is even and j is odd; c) (a i , µ i ) = (−a j , −µ j ), i is odd and j is even; d) (a i , µ i ) = (−a j , −µ j ), i is even and j is odd.Note that the first two cases can occur when a i = a j ∈ E(Ω(A)) while the latter two cases when a i = −a j ∈ ±E(Θ(A)).
CASE a).Let us assume (a i , µ i ) = (a j , µ j ) where i is odd and j is even.Then and hence we have that . On the other hand, we have that Also, since a j ∈ E(Ω(A)), µ j = −µ j−1 and hence Note that this is a contradiction since we are assuming that i and j are at a minimal distance and j − i > 1.
CASE b).Let us assume (a i , µ i ) = (a j , µ j ) where i is even and j is odd.Then, On the other hand we have that Also, since a j ∈ E(Ω(A)), µ j = −µ j−1 and hence a j−1 µ j−1 = µ j λ(ω µj r (a j ))ω µj r (a j ).Finally from (a j , µ j ) = (a i , µ i ), it follows that: Note that this is a contradiction since we are assuming that i and j are at a minimal distance and j − i > 1. CASE c).Let us assume (a i , µ i ) = (−a j , −µ j ) where i is odd and j is even.Then . On the other hand, we have that Also, since a j ∈ ±E(Θ(A)), µ j = µ j−1 and hence , it follows that: Note that this is a contradiction since we are assuming that i and j are at a minimal distance and j − i > 1.
CASE d).Let us assume (a i , µ i ) = (−a j , −µ j ) where i is even and j is odd.Then, and λ(a i ) = 1, we have that a i+1 µ i+1 = µ i λ(ω µi r (−a i ))ω µi r (−a i ).On the other hand we have that Also, since a j ∈ ±E(Θ(A)), µ j = µ j−1 and hence Finally from (−a j , −µ j ) = (a i , µ i ), it follows that: Note that this is a contradiction since we are assuming that i and j are at a minimal distance and j − i > 1.
Therefore a i µ i = a j µ j can hold only assuming j ≡ i (mod 2) and hence, as noted above, (a i , µ i ) = (a j , µ j ).It follows that (a i+ℓ , µ i+ℓ ) = (a j+ℓ , µ j+ℓ ) for any positive integer ℓ, which means that T |(j − i) where T = 2nk is the period of ((a i , µ i )), but this is in contradiction with the assumption that 2nk ≥ j > i ≥ 1.Thus, assuming (1), i.e.T = 2nk, the first 2nk elements of (a i µ i ) are distinct.
(2) ⇒ (3).Assuming that (2) holds, the first 2nk elements of (a i µ i ) are different.Hence the period T ′ of (a i µ i ) is larger than or equal to 2nk.Let us consider the element a 2nk+1 µ 2nk+1 .Since it is a nonzero element of Z 2nk+1 , it must be equal to a j µ j for some j ∈ [1, 2nk].Moreover, due to the previous discussion, a 2nk+1 µ 2nk+1 = a j µ j can occur only if j ≡ 2nk + 1 (mod 2) and (a i , µ i ) = (a j , µ j ) which implies that T |(2nk + 1 − j) where T is the period of ((a i , µ i )).Note that T is a multiple of the period T ′ of the sequence (a i µ i ).It follows that This can occur only if j = 1 and if the period of (a i µ i ) is exactly 2nk.Therefore also property (3) holds.
(4) ⇒ (1).Since the period T ′ of (a 2i+1 , µ 2i+1 ) is nk, then the period T of (a i , µ i ) must be a multiple of 2nk.Moreover, since (a 2nk+1 , µ 2nk+1 ) = (a 1 , µ 1 ), and 1 ≡ 2nk + 1 (mod 2), we have that (a 2nk+1+ℓ , µ 2nk+1+ℓ ) = (a 1+ℓ , µ 1+ℓ ) for any positive integer ℓ.Therefore, the period of (a i , µ i ) must be exactly 2nk.Now we can use the sequence ((a i , µ i )) in order to generalize the definition of compatible orderings given in [1].Definition 5.7.Let A be a WH(m, n; h, k) and consider the ordering ω r for the rows and ω c for the columns.Then we say that ω r and ω c are compatible whenever the conditions of Proposition 5.6 are satisfied.Definition 5.8 (Archdeacon embedding).Let A be a WH(m, n; h, k) that admits compatible orderings ω r and ω c .According to Proposition 5.6, the following list can be seen as a cyclic permutation of Z 2nk+1 \ {0} Then we define the map ρ on the set of the oriented edges of the complete graph K 2nk+1 as follows ρ((x, x + a)) = (x, x + ρ 0 (a)).(5.4) Clearly, given compatible orderings ω r and ω c , the map ρ is a rotation of K 2nk+1 .
Archdeacon [1] proved that the following theorem holds.
Theorem 5.9.Let A be a WH(m, n; h, k) that admits two compatible orderings ω r and ω c .Then there exists a cellular biembedding ψ of K 2nk+1 , such that every edge is on a face whose boundary length is h and on a face whose boundary length is k.Moreover, ψ is Z 2nk+1 -regular.
We are also interested in describing the faces (and their lengths) induced by the Archdeacon embedding under the condition of Theorem 5.9.
For this purpose, we take a WH(m, n; h, k) that admits two compatible orderings ω r and ω c .Since an embedding of Archdeacon type can be seen as a special case of derived embedding from current graphs (see [1] and [12]), given a ∈ E(R) for a suitable row R, the oriented edge (x, x + a) belongs to the face F 1 whose boundary is (5.6) x, x + a, x + a + ω r (a), . . ., x Let us now consider the oriented edge (x, x + a) with −a ∈ E(C) for a suitable column C. Then (x, x + a) belongs to the face F 2 whose boundary is while (µ 1 , . . ., µ 24 ) is It follows that the rotation ρ 0 is given by the cycle: Finally, in order to define the embedding, we need also to define the map ǫ : E(K 2nk+1 ) → {−1, 1}.According to Equation (5.5), this map acts as follows:

An infinite class of non-orientable embeddings
Embeddings of Archdeacon type into orientable surfaces have been considered by several papers (see, for instance, [5,8,9,10]) but, apart from a single example exhibited in [1], no one has investigated such embeddings in the non-orientable case yet.For this reason, in this section, we aim to present an infinite family of nonorientable embeddings of Archdeacon type which arise from weak Heffter arrays.6.1.Crazy Knight's Tour Problem.Proposition 5.6 leads us to consider the following problem which can be seen as a generalization of the original Crazy Knight's Tour Problem proposed in [6].Given a WH(m, n; h, k), say A, by r i we denote the orientation of the i-th row, 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 the orientations R = (r 1 , . . ., r m ) and C = (c 1 , . . ., c n ) are fixed.Now we consider the following tour on two identical copies of the array A that we denote by A 1 and by A −1 .More precisely, we first denote by Skel(A) the set of non-empty cells of A. Then we index the nonempty cells of A 1 with the triples (i, j, 1) where (i, j) ∈ Skel(A).Similarly, we index the nonempty cells of A −1 using the triples (i, j, −1) where (i, j) ∈ Skel(A).Finally, given a cell (i, j, t) in the array A t (here t is either 1 or −1) we consider the moves: 1) L R (i, j, t) is the cell (i, j ′ , t ′ ) where j ′ is the column index of the filled cell of the row R i next to (i, j) in the orientation r t i and t ′ = t if (i, j ′ ) ∈ Skel(Θ(A)) and t ′ = −t if (i, j ′ ) ∈ Skel(Ω(A)).2) L C (i, j, t) is the cell (i ′ , j, t ′ ) where i ′ is the row index of the filled cell of the column C j next to (i, j) in the orientation c t j and t ′ = t if (i ′ , j) ∈ Skel(Θ(A)) and t ′ = −t if (i, j ′ ) ∈ Skel(Ω(A)).Then, assuming (1, 1) ∈ Skel(A) and setting we consider the list where ℓ is the minimum value such that (L R •L C ) ℓ+1 (1, 1, t) = (1, 1, t).The problem we propose here is the following.
Crazy Knight's Tour Problem.Given a weak Heffter array A, do there exist C and R such that the list L C,R has length |Skel(A)|?
Clearly the Crazy Knight's Tour Problem can be stated more in general for a partially filled array A and it is denoted by P (A).Here, if L C,R has length |Skel(A)| we say that (C, R) is a solution of P (A).Example 6.1.Let A be again the WH (3,4) of Example 1.3.Since the arrays A 1 and A −1 are copies of A, we have that Now we consider the orientations C = (−1, −1, 1, 1) and R = (1, 1, 1).Then the list L C,R is given by: ( We can represent the orientations and the tour directly on the arrays A 1 and A −1 as follows: here the arrows represent the orientations (C, R on A 1 and their opposites on A −1 ), we have highlighted in grey the cells of Skel(Ω(A)) and the numbers represent the positions of the cells in the tour.
The relationship between the Crazy Knight's Tour Problem and Archdeacon embeddings is explained by the following remark.Remark 6.2.If A is a WH(m, n; h, k) such that P (A) admits a solution (C, R), then A also admits two compatible orderings ω c and ω r that can be determined as follows.For each row R (resp.column C), we consider ω R to be the natural ordering if r i = 1 (resp.c i = 1) and its opposite otherwise.As usual we set We consider now the sequence (a i ) defined as in Equation (5.1) starting from the orderings ω c , ω r associated with the orientations C and R. Because of the definition of L C,R , the element (i, j, t) in the ℓ-th position of this list is such that the cell (i, j) of A, considered with its row sign (i.e. its upper sign), contains a 2ℓ+1 and µ 2ℓ+1 = t.It follows that, if the list L C,R has length |Skel(A)| = nk, then the sequence ((a 2i+1 , µ 2i+1 )) has period nk.But, due to Proposition 5.6, this means that ω r and ω c are compatible.
A consequence of this remark is that solutions of P (A) provide embeddings of complete graphs, more precisely: Theorem 6.3.Let A be a WH(m, n; h, k) such that P (A) admits a solution (C, R).Then there exists a cellular biembedding ψ of K 2nk+1 , such that every edge is on a face whose boundary length is a multiple of h and on a face whose boundary length is a multiple of k.Moreover, ψ is Z 2nk+1 -regular.
Note that the embedding obtained from a solution (C, R) of P (A) via Theorem 6.3 is exactly the Archdeacon embedding of K 2nk+1 introduced in Definition 5.8 starting from the compatible orderings ω c , ω r associated to (C, R).

Conditions of non-orientability.
In this subsection we present infinitely many non-orientable embeddings of Archdeacon type.For this purpose, we recall a general orientability criterion (see [12]).Theorem 6.4.A combinatorial embedding Π = (Γ, ǫ, ρ) is orientable if and only if any cycle C of Γ contains an even number of edges e such that ǫ(e) = −1 (in the following, edges of type 1).
As a consequence, we can state here the following orientability criterion for embeddings of Archdeacon type.Theorem 6.5.Let Π = (K 2nk+1 , ǫ, ρ) be an embedding of Archdeacon type.Then Π is orientable if and only if ǫ(e) = 1 for all e ∈ E(K 2nk+1 ) (in the following, edges of type 0).
Proof.Let us suppose that every edge of K 2nk+1 is of type 0.Then, any cycle of K 2nk+1 contains zero edges of type 1.Hence, by Theorem 6.4, the embedding Π is orientable.Now, suppose that there is an edge, say (x, x + a), of type 1.Then, since the type of (x, x + a) is independent from the value of x, we have that also the edges (x + a, x + 2a), (x + 2a, x + 3a), . . .are of type 1. Denoted by ℓ the additive order of a in Z 2nk+1 , since |Z 2nk+1 | is odd, we have that also ℓ is odd.It follows that the cycle C := (x, x + a, x + 2a, . . ., x + (ℓ − 1)a) contains an odd number (i.e.ℓ) of edges of type 1.Hence, because of Theorem 6.4, Π is non-orientable.
Now we can state a non-orientability condition that refers to the problem P (A).
Proposition 6.6.Let A be a WH(m, n; h, k), let (C, R) be a solution of P (A) and let Π = (K 2nk+1 , ǫ, ρ) be the associated Archdeacon embedding.If there exists (i, j) such that both (i, j, 1) and (i, j, −1) belong to L C,R , then the embedding Π is non-orientable.
Proof.Let us suppose that both (i, j, 1) and (i, j, −1) appear in the list L C,R .We may assume that (i, j, 1) is in the ℓ-th position of this list and (i, j, −1) is in the ℓ ′ -th one.Now we consider the sequence (a i ) defined as in Equation (5.1) with the orderings associated with the orientations C and R. Because of the definition of L C,R and since (i, j, 1) is the ℓ-th element of this list, the element in the cell (i, j) of A considered with its row sign (i.e. its upper sign) is a 2ℓ+1 .On the other hand, since (i, j, −1) is the ℓ ′ -th element in the list L C,R , the element in the cell (i, j) of A considered with its row sign is also a 2ℓ ′ +1 .Therefore, we have at least one repetition in the sequence (a 2i+1 ) and hence, because of Definition 5.8, there exists at least one edge e such that ǫ(e) = −1.It follows from Theorem 6.5 that the embedding Π is not orientable.Since both (1, 1, −1) and (1, 1, 1) appear in L C,R , it follows from Proposition 6.6 that this solution of P (A) induces a non-orientable embedding of K 25 of Archdeacon type.
It follows from Proposition 6.6 that this solution of P (A) induces a non-orientable embedding of K 6n+1 of Archdeacon type.

Conclusions
We have several values of the parameters n, k, t for which there exists an H t (n; k), but not a strictly weak WH t (n; k), see Remark 3.1, and Propositions 3.9 and 3.10.Clearly, there is no reason to believe that these are the unique choices of the parameters for which this happens.On the other hand, we have no example about the existence of a strictly weak WH t (n; k) when an H t (n; k) does not exist.By the way, we emphasize that in Theorem 4.2 we provide the existence of a strictly weak WH 5 (n; 5) for every n ≡ 0 (mod 4) with n ≥ 12 since this case was left open in [7].Hence, up to now, the existence of an H 5 (n; 5) for n ≡ 0 (mod 4) is unknown, except for n = 8, 16, [7].For these arguments, we believe that the following question naturally arises: if there exists a strictly weak WH t (n; k) does then an H t (n; k) exist too?At the moment we do not see any reasons for which the existence of a strictly weak Heffter array implies that of a classical one.But, as already remarked above, we have no example which allow us to give a negative answer to previous question.Hence we propose the following.

Theorem 4 . 1 .
Let 3 ≤ k ≤ n with k = 5.There exists an integer H k (n; k) if and only if one of the following holds:
[7]ce for all these values of the parameters neither a classical nor a strictly weak Heffter array can exist.(1)Theexistence of an H t (4; 3) is known for t = 1, see Theorem 1.4, and t = 3, see Theorem 1.5 of[7].The following are an H 2 (4; 3), an H 4 (4; 3) and an H 6 (4; 3): 4n + 1].Now starting from B we construct a new array C by adding to B the arrays M i in the following positions.The elements of the array M 1 are inserted in the positions (1, n + 1).Finally, for all i ∈ { n 2 + 3, . . ., n − 1} the elements of the array ±M T i , where by M T i we mean the transpose of M i , are inserted in the positions ( [7] + 3, n 2 + 5, ..., n − 1} we exchange the element C[i, i] with C[i+1, i+1].Finally, in the position ( n Clearly, the support of this new array is nothing but the support of C. The property that now the total sum of the rows and the columns is zero follows by combining these last assignments with Equation (4.1).Example 4.3.In this example we follow step by step the proof of Theorem 4.2 to construct a strictly WH 5(12; 5).Firstly, let A be the H 3 (12; 3) obtained by applying the construction of Proposition 5.3 of[7](for convenience we have highlighted its subdivision into 2 × 2 blocks):We then insert the elements of these arrays in B obtaining the new array C: