ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 17 (2019) 369-395 https://doi.org/10.26493/1855-3974.1817.b97 (Also available at http://amc-journal.eu) The existence of square non-integer Heffter arrays Nicholas J. Cavenagh Department of Mathematics, The University of Waikato, Private Bag 3105, Hamilton 3240, New Zealand Jeffery H. Dinitz Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05405, USA Diane M. Donovan School of Mathematics and Physics, The University of Queensland, Queensland 4072, Australia Emine §ule Yazici Department of Mathematics, Koc University, Istanbul, Turkey Received 8 October 2018, accepted 13 April 2019, published online 4 November 2019 A Heffter array H(n; k) is an n x n matrix such that each row and column contains k filled cells, each row and column sum is divisible by 2nk + 1 and either x or —x appears in the array for each integer 1 < x < nk. Heffter arrays are useful for embedding the graph K2nk+1 on an orientable surface. An integer Heffter array is one in which each row and column sum is 0. Necessary and sufficient conditions (on n and k) for the existence of an integer Heffter array H(n; k) were verified by Archdeacon, Dinitz, Donovan and Yazici (2015) and Dinitz and Wanless (2017). In this paper we consider square Heffter arrays that are not necessarily integer. We show that such Heffter arrays exist whenever 3 < k < n. Keywords: Heffter arrays, biembedding cycle systems. Math. Subj. Class.: 05B30 E-mail addresses: nickc@waikato.ac.nz (Nicholas J. Cavenagh), jeff.dinitz@uvm.edu (Jeffery H. Dinitz), dmd@maths.uq.edu.au (Diane M. Donovan), eyazici@ku.edu.tr (Emine Sule Yazici) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/4.G/ 370 Ars Math. Contemp. 17 (2019) 349-368 1 Introduction A Heffter array H(m, n; s, t) is an m x n matrix of integers such that: (1) each row contains s filled cells and each column contains t filled cells; (2) the elements in every row and column sum to 0 in Z2ms+1; and (3) for each integer 1 < x < ms, either x or — x appears in the array. If the Heffter array is square, then m = n and necessarily s = t. We denote such Heffter arrays by H(n; k), where each row and each column contains k filled cells. A Heffter array is called an integer Heffter array if Condition (2) in the definition of a Heffter array above is strengthened so that the elements in every row and every column sum to zero in Z. Archdeacon, in [1], was the first to define and study a Heffter array H(m, n; s, t). He showed that a Heffter array with a pair of special orderings can be used to construct an embedding of the complete graph K2ms+1 on a surface. This connection is formalised in the following theorem. For definitions of simple and compatible orderings refer to [1]. Theorem 1.1 ([1]). Given a Heffter array H(m,n; s,t) with compatible orderings ur of the symbols in the rows of the array and on the symbols in the columns of the array, then there exists an embedding of K2ms+1 such that every edge is on a face of size s and a face of size t. Moreover, if wr and wc are both simple, then all faces are simple cycles. The embedding of K2ms+1 given in Theorem 1.1 provides a connection with the embedding of cycle systems. A t-cycle system on n points is a decomposition of the edges of Kn into t-cycles. A t-cycle system C on Kn is cyclic if there is a labeling of the vertex set of Kn with the elements of Zn such that the permutation x ^ x + 1 preserves the cycles of C. A biembedding of an s-cycle system and a t-cycle system is a face 2-colorable topological embedding of the complete graph K2ms+1 in which one color class is comprised of the cycles in the s-cycle system and the other class contains the cycles in the t-cycle system, see for instance [4, 5, 6, 8, 9, 10, 11] for further details. A number of papers have appeared on the construction of Heffter arrays, H(m, n; s, t). The case where the array contained no empty cells was studied in [2], with results summarised in Theorem 1.2. Theorem 1.2 ([2]). There is an H(m, n; n, m) for all m,n > 3 and an integer Heffter array H (m, n; n, m) exists if and only if m, n ^ 3 and mn = 0, 3 (mod 4). The papers [3, 7] focused on square integer Heffter arrays H(n; k) and verified their existence for all admissible orders. This result is summarized in the following theorem. Theorem 1.3 ([3, 7]). There exists an integer H(n; k) if and only if 3 < k < n and nk = 0, 3 (mod 4). Table 1 lists the possible cases and cites the article which verifies existence of square integer Heffter arrays, where DNE represents a value that does not exist. In these cases we will verify existence for the non-integer Heffter arrays H(n; k). The main result of this paper is the following. Theorem 1.4. There exists an H (n; k) if and only if 3 < k < n. N. J. Cavenagh et al.: The existence of square non-integer Heffter arrays 371 Table 1: Existence results for square integer Heffter arrays H(n; k). n \ k 0 1 2 3 0 [3] [3,7] [3] [3] 1 [3] DNE DNE [3] 2 [3] DNE [3] DNE 3 [3] [3,7] DNE DNE Table 2: Cases for non-integer Heffter arrays H(n; k). Case A Case B CaseC CaseD Case E k 2 (mod 4) 3 (mod 4) 3 (mod 4) 1 (mod 4) 1 (mod 4) n 1, 3 (mod 4) 3 (mod 4) 2 (mod 4) 1 (mod 4) 2 (mod 4) From Theorem 1.2 above, the case n = k has been solved, so we henceforth assume that n > k. The cases that need to be addressed are set out in Table 2. Cases A, B, C, D and E are solved by Theorems 3.2,4.2, 5.2,6.2 and 7.2, respectively, thus proving Theorem 1.4. In this paper the rows and columns of a square n x n array are always indexed by the elements of {1, 2,..., n}. Unless otherwise stated, when working modulo n, replace 0 by n, so we use the symbols 1, 2,..., n instead of 0,1,...,n -1. While rows and columns are calculated modulo an integer, entries are always expressed as non-zero integers. Throughout this paper A[r, c] = x denotes the occurrence of symbol x in cell (r, c) of array A. By A±z we refer to the array obtained by replacing A[r, c] by A[r, c]+z (if A[r, c] > 0) and A[r, c] -z (if A[r, c] < 0). If each row and each column of A contains the same number of positive and negative numbers, then A ± z has the same row and column sums as A. In this case we say A is shiftable. The support of an array A is defined to be the set containing the absolute value of the elements contained in A. If A is an array with support S and z a nonnegative integer, then A ± z has support S + z. 2 Increasing k from base cases For each of the cases set out in Table 2 our overall strategy is to generate a base case H(n; k) where k takes the smallest possible value and then increase k by multiples of 4, adjoining 4 additional entries to each row and column. In this section we outline various tools to enable this process. To this end, we introduce the following definitions. We associate the cells of an n x n array with the complete bipartite graph Kn,n where partite sets are denoted {aj | i = 1, 2,..., n} and {bj | j = 1,2,..., n} and the edge {aj, bj} corresponds to the cell (i, j). We say that in an n x n array a set of cells S forms a 2-factor if the corresponding set of edges in the graph Kn,n forms a spanning 2-regular graph and forms a Hamilton cycle if the corresponding set of edges forms a single cycle of length 2n. For each d G {0, 1,...,n - 1},we define the diagonal Dd to be the set of cells of the form (r + d, r), 1 < r < n (evaluated modulo n). Observe that the cells Dj U Dj form a Hamilton cycle whenever j - i is coprime to n. Lemma 2.1. Let Si and S2 be two disjoint sets of cells in an n x n array which each form 372 Ars Math. Contemp. 17 (2019) 349-368 Hamilton cycles. The cells of Si U S2 can be filled with the elements of {1,2,..., 4n} so that each row and column sum is equal to 8n + 2. Proof. Let the cells of Si and S2 be {ei | 1 < i < 2n} and {fi | 1 < i < 2n}, respectively, where: • Cells ei and ei+i are in the same row (column) whenever i is odd (respectively, even); • Cells fi and fi+i are in the same row (column) whenever i is odd (respectively, even); • Cells ei and fi are in the same row. Place 1 in cell ei, 4n in cell fi and: • 2n — 2i + 1 in cell e2i+i, where 1 < i < n — 1; 2n + 2i — 1 in cell e2i where 1 ^ i ^ n; • 2n + 2i in cell f2i+i where 1 < i < n — 1; 2n — 2i + 2 in cell f2i where 1 < i < n. The entries in cells ei, e2, fi and f2 add to 1 + (2n + 1) + 4n + 2n = 8n + 2. For every other row, there are two cells from Si with entries adding to 4n + 2 and two cells from S2 with entries adding to 4n. For every column, there are two cells from Si adding to 4n and two cells from S2 adding to 4n + 2. See the example below. □ We demonstrate Lemma 2.1 below when n = 9. The elements of Si are underlined. Si U S2 1 19 18 36 34 17 21 2 15 23 10 26 4 13 25 32 14 11 27 22 20 9 29 16 30 7 31 6 24 12 5 33 35 8 28 3 The following theorem will be crucial in Cases A and D. Theorem 2.2. Let H(n; k) be a Heffter array such that each row and column sums to 2nk + 1. Suppose there exist Hamilton cycles Hi and H2 disjoint to each other and to the filled cells of H(n; k). Then there exists an H(n; k + 4) Heffter array with row and column sums equal to 2n(k + 4) + 1, where the filled cells are precisely the filled cells of H (n; k), Hi and H2. Proof. Let A0 represent the H(n; k) and negate each element so that each row and column has sum equal to —(2nk + 1). From Lemma 2.1, there exists an array Ai on the cells of Hi and H2 such that each row and column sum is equal to 8n + 2; add n(k + 4) — (4n) = nk to N. J. Cavenagh et al.: The existence of square non-integer Heffter arrays 373 each element of Ai to create a new array A i that has support { nk +1, nk+2,..., n (k+4)}. Note that in Ai each row and column sum is equal to 8n + 2 + 4nk. Let A be the union of A0 with Ai. The resulting array A has support {1, 2,..., n(k + 4)}, with k + 4 filled cells in each row and column. Finally, each row and column sum of A is — (2nk + 1) + (8n + 2) + 4(nk) = 2n(k + 4) + 1, as desired. □ The following lemma generalizes Theorem 2.2 from [7], and is used in Cases B and C. Lemma 2.3. Let Si and S2 be two disjoint sets of cells in an n x n array which each form Hamilton cycles. Then for any positive integers t and s > t + 2n, the cells of Si U S2 can be filled with elements to make a shiftable array with support {s + i, t + i | 1 < i < 2n} so that the four elements in each row and each column sum to 0. Proof. Let the sets of cells of Si and S2 be {ej | 1 < i < 2n} and {/ | 1 < i < 2n}, respectively, where: • Cells ei and ei+i are in the same row (column) whenever i is odd (respectively, even); • Cells fi and fi+i are in the same row (column) whenever i is odd (respectively, even); • Cells ei and fi are in the same row. Place: • s + 2n in cell ei and — (t + 2n) in cell fi, with sum s — t; • s + 2i in cell e2i+i and —(t + 2i) in cell f2i+i, with sum s — t, for 1 < i < n — 1, • —(s + 2i — 1) in cell e2i and t + 2i — 1 in cell f2i, with sum t — s, for 1 < i < n. It now follows that the row sums are 0. Using similar arguments it can be seen that the columns also sum to 0. Observe that there are two positive and two negative integers in each row and column; thus the array is shiftable. □ The proof of the following lemma is similar to the proof of Lemma 2.3; we use this in Case E. Lemma 2.4. Let n be even. Let Si and S2 be two disjoint sets of cells in an n x n array which each form 2-factors that are the union of two n-cycles. Then for any positive integers s, t, u and v where s>t + n, t > u + n and u > v + n, the cells of Si U S2 can be filled with elements to make a shiftable array with support {s + i,t + i, u + i,v + i | 1 < i < n} so that the four elements in each row and column sum to 0. Proof. Let Ci, Cj be the cycles of the 2-factor Sj, i € {1, 2}, where Ci and Ci share a row and C2 and C2 share a row. Let the sets of cells of Ci, C{, C2 and C2 be {ei | 1 < i < n}, {/i | 1 ^ i ^ n}, {gi | 1 < i < n} and {hi | 1 < i < n}, respectively, where: • Cells ei and ei+i are in the same row (column) whenever i is odd (respectively, even); 374 Ars Math. Contemp. 17 (2019) 349-368 • Cells fi and fi+1 are in the same row (column) whenever i is odd (respectively, even); • Cells gi and gi+1 are in the same row (column) whenever i is odd (respectively, even); • Cells hi and hi+1 are in the same row (column) whenever i is odd (respectively, even); • Cells e1 and g1 are in the same row; cells f1 and h1 are in the same row. Place: • s + n in cell e1, — (t + n) in cell g1 and u + n in cell f1, — (v + n) in cell h1; • s + 2i in cell e2i+1 and — (t + 2i) in cell g2i+1, for 1 < i < n/2 — 1; • —(s + 2i — 1) in cell e2i and t + 2i — 1 in cell g2i, for 1 < i < n/2; • u + 2i in cell f2i+1 and —(v + 2i) in cell h2i+1, for 1 < i < n/2 — 1; • —(u + 2i — 1) in cell f2i and v + 2i — 1 in cell h2i, for 1 < i < n/2. It now follows that the row sums are 0. Using similar arguments it can be seen that the columns also sum to 0. □ 3 Case A: k = 2 (mod 4) In this section we construct a Heffter array H(n; k), for n = 1,3 (mod 4) and k = 2 (mod 4), where k < n. Row and column sums will always equal 2nk + 1. We start with an example of our construction. H(15; 6) 6 —4 89 81 1 8 12 —88 83 87 85 2 86 18 —82 77 3 79 73 80 24 —76 71 9 15 67 74 30 —70 65 59 21 61 68 36 —64 —58 53 27 55 62 42 —52 47 33 49 56 48 —46 41 39 43 50 54 —40 35 45 37 44 60 —34 29 51 31 38 66 —28 23 57 25 32 72 —22 17 63 19 26 78 — 16 11 69 13 20 84 — 10 5 75 7 14 90 N. J. Cavenagh et al.: The existence of square non-integer Heffter arrays 375 Lemma 3.1. For n = 1, 3 (mod 4), n ^ 7 and k = 6 there exists a Heffter array H(n; 6). Proof. We remind the reader that rows and columns are calculated modulo n but the array entries are not. The array A = A[r, c] is defined as follows, where 1 < i < n: A[i,i] = 6i, A[i + 2, i] = 6n + 2 - 6i, A[i + 1, n - 2 + i] = 6n +1 - 6i, A[i + 2, n - 2 + i] = 6i - 3, A[i, n - 5 + i] = 6n + 5 - 6i, A[i + 1, n - 5 + i] = -6n - 4 + 6i. Then the support of A is {1, 2,..., 6n}. The sets of elements in rows 1, 2 and i, 3 < i < n, are, respectively: {6, 8,1, 6n - 9, 6n - 1, -4}, {12, 2, 6n - 5, 6n - 3, 6n - 7, -(6n - 2)}, {6i, 6n + 2 - 6(i - 2), 6n +1 - 6(i - 1), 6(i - 2) - 3, 6n + 5 - 6i, -6n - 4 + 6(i - 1)}. Thus in each case the sum of elements in a row is 12n +1. The set of elements in column i, 1 < i < n - 5 is: {6i, 6n + 2 - 6i, 6n +1 - 6(i + 2), 6(i + 2) - 3, 6n + 5 - 6(i + 5), -6n - 4 + 6(i + 5)}. The set of elements in columns n - 4, n - 3, n - 2, n - 1 and n are, respectively: {6n - 24, 26,13, 6n - 15, 6n - 1, -(6n - 2)}, {6n - 18, 20, 7, 6n - 9, 6n - 7, -(6n - 8)}, {6n - 12,14,1, 6n - 3, 6n - 13, -(6n - 14)}, {6n - 6, 8, 6n - 5, 3, 6n - 19, -(6n - 20)}, {6n, 2, 6n - 11, 9, 6n - 25, -(6n - 26)}. Thus in each case the sum of elements in a column is 12n +1. □ Theorem 3.2. There exists a Heffter array H(n; k) for all n = 1, 3 (mod 4) and k = 2 (mod 4), where n > k ^ 6. Proof. Let k = 4p + 6. Then 4p + 6 < n - 1 so p < (n - 7)/4. We have solved the case p = 0 in Lemma 3.1 so we may assume p > 1. Observe that the Heffter array given in the proof of that lemma uses only elements in diagonals D0, D2, D3, D4, D5 and D6 and so does not intersect the diagonals D7, D8,..., Dn-1. We can apply Theorem 2.2 recursively, where the diagonals D7, D8,..., D6+2p-1, D6+2p can be paired to give sets of cells S1 and D6+2p+1, D6+2p+2,..., D6+4p-1, D6+4p paired to give sets of cells S2. The result is a Heffter array H(n; k) with constant row and column sum 2nk + 1 whenever k = 2 (mod 4), n = 1, 3 (mod 4) and n > k > 6. □ 4 Case B: k = 3 (mod 4) and n = 3 (mod 4) In this section we construct a Heffter array H(n; k) where n = 4m + 3, k = 4p + 3 and k < n. 376 Ars Math. Contemp. 17 (2019) 349-368 We will begin with k = 3. We first assume that m > 4 and construct an n x n array which is the concatenation of three smaller arrays, AO = AO[r, c], of dimension (4m — 7) x (4m — 7), A1 = A^r, c] of dimension 7 x 7 and C of dimension 3 x 3, each containing 3 filled cells per row and column. So we see that n = 4m + 3. The sum of the rows and columns in AO and A1 will be 0, while the sum of the rows and columns in C will be 2nk + 1. We begin with an example of the main construction of this section. H (19; 3) 16 -48 32 17 27 -44 -33 -14 47 21 15 -36 -18 -13 31 29 -9 -20 -34 -12 46 45 -10 -35 -19 30 -11 -25 24 1 22 -50 28 3 37 -40 26 23 -49 -38 42 -4 -51 8 43 -2 41 -39 5 57 53 54 6 55 56 52 7 Lemma 4.1. For n = 3 (mod 4) and n ^ 7 there exists a Heffter array H(n; 3). Proof. Let n = 4m + 3. We first assume that m > 4 and so n > 19. The small cases will be dealt with at the end of the proof. Let AO be: A0[2i — 1, 2i] = 8m + 1 — i, 1 < i < m, AO[2i, 2i — 1] = —(8m + i), 1 < i < m, AO [2i, 2i + 1] = 12m — i, 1 < i < m — 1, AO[2i + 1, 2i] = —(4m + 1 + i), 1 < i < m — 1, AO [2m — 2 + 2i, 2m — 1 + 2i] = 5m + i, 1 < i < m — 3, AO[2m — 1 + 2i, 2m — 2 + 2i] = —(11m + 1 — i), 1 < i < m — 3, N. J. Cavenagh et al.: The existence of square non-integer Heffter arrays 377 A0 [2m - 1 + 2«, 2m + 2i] = 9m + i, 1 < i < m - 4, A0[2m + 2i, 2m - 1 + 2i] = -(7m +1 - i), 1 < i < m - 4, A0[i + 1,i + 1] = -(4m - 1 - i), 1 < i < 2m - 2, A0 [2m + i, 2m + i] = 2m - i, 1 < i < 2m - 8, A0[2m, 2m] = 4m - 1, A0[1,4m - 7] = -12m, A0[1,1] = 4m, A0 [4m - 7,4m - 7] = 6m + 3. A0[4m - 7,1] = 4m + 1, We illustrate Ai, in the case m — 4: A0(m = 4) 16 32 -48 -33 -14 47 -18 -13 31 -34 -12 46 -19 -11 30 -35 -10 45 -20 -9 29 -36 15 21 17 -44 27 First observe that the array A0 is a (4m - 7) x (4m - 7) array that has 3 filled cells in each row and column. To confirm that the row and columns sums are 0, note that this array was constructed by taking the first 4m - 8 rows and columns of the integer Heffter array H(4m; 3) given in [3], then placing entry -12m in cell (1,4m - 7), entry 4m +1 in cell (4m - 7,1) and entry 6m + 3 in cell (4m - 7,4m - 7). Thus we need only check the sum of row 4m - 7 which is (4m + 1) + (6m + 3) - (10m + 4) = 0 and the sum of column 4m - 7 which is -12m + (6m - 3) + (6m + 3) = 0. Hence each row and column in the array A0 sums to zero. Although not necessary for the case k = 3, for larger values of k (see the following theorem) we map the rows and columns of A0 so that the filled cells are a subset of the union of diagonals D0 U D1 U D2 U Dn-2 U Dn-1. This can be done by applying the mapping P 2i - 1, when 1 < i < 2m - 3, i 1 8m - 2i - 12, when 2m - 2 < i < 4m - 7, to the rows and column of A0. This does not change the row and column sum and the support is still {1, 2,.. ., 4m + 3} \ {1, 2, .. ., 7, 2m, 6m - 2,. .., 6m + 2, 6m + 4, 10m - 3, .. ., 10m + 3, 12m + 1,12m + 2,. .., 12m + 9}. We call this rearranged array A0; see H(19; 3) above for A0 when m = 4. 378 Ars Math. Contemp. 17 (2019) 349-368 Next, let Ai be: A i — (6m + 1) 6m 1 6m — 2 — (12m + 2) 6m + 4 3 10m — 3 — 10m 6m + 2 6m — 1 — (12m + 1) — (10m — 2) 10m + 2 —4 — (12m + 3) 2m 10m + 3 —2 10m + 1 — (10m — 1) It is easy to check that this array has row and column sum 0 and support {1, 2, 3,4, 2m, 6m - 2,..., 6m + 2, 6m + 4, 10m - 3, .. ., 10m + 3, 12m + 1,12m + 2,12m + 3}. We place A1 on the intersection of row and column sets {4m — 6,4m — 5,..., 4m}. Finally, place the block C on the intersection of the row and column sets {4m + 1, 4m + 2,4m + 3}. Observe that the rows and columns sum to 2nk + 1. It is convenient to express C in terms of n and k, as it will be part of more general constructions in the next theorems. However, if k = 3, observe that {12m + 4,12m + 5,..., 12m + 9} = {nk — 5, nk — 4,. .., nk}. C 5 nk nk — 4 nk — 3 6 nk — 2 nk — 1 nk — 5 7 Let A be the concatenation of A0, A1 and C to obtain an H(4m + 3; 3) Heffter array for all m > 4. An H(7; 3) is given in the Appendix. When n = 11 or 15, concatenating the array B(n) given in the Appendix for these size with C will produce an H(n; 3) with similar properties and support. □ We now consider the case when k > 3; we apply the techniques developed in Section 2. Theorem 4.2. There exists a Heffter array H(n; k) for all n = 3 (mod 4) and k = 3 (mod 4), where n > k ^ 3. Proof. Given Lemma 4.1 we only need to address the case k = 4p + 3 where 1 < p < m. Since k < n — 4, p < (n — 7)/4. First observe that the filled cells of H(n; 3) given in the previous lemma are a subset the union of diagonals D0, D1, D2, Dn-2, Dn-1 with support {1, 2,..., 12m+3}U{nk—5, nk—4,..., nk}. We next identify p disjoint Hamilton cycles, that are also disjoint from diagonals D0, D1, D2, Dn-2, Dn-1, by pairing the remaining (n —5) diagonals. Then we apply Lemma 2.3, contributing {12m+4,12m+5,... ,nk — 6} to the support. The result is a Heffter array H(n; k) with each row and column sum equal to 0 except for the final three rows and columns which sum to 2nk + 1. □ N. J. Cavenagh et al.: The existence of square non-integer Heffter arrays 379 5 Case C: k = 3 (mod 4) and n = 2 (mod 4) We construct a Heffter array H(n; k) where n = 4m + 6 and k = 4p + 3 for k < n. As in the previous section, we will begin with k = 3. Firstly we will assume that m > 7 and construct an n x n array which is the concatenation of several smaller arrays. Here we use four smaller arrays, AO = AO[r, c] of size (4m - 13) x (4m - 13), A1 = A^r, c] of size 13 x 13, A2 = A2 [r, c] of size 3 x 3 and A3 = A3[r, c] of size 3 x 3, each containing 3 filled cells per row and column. The sum of the rows and columns in A0, A1 and A2 is 0, while the sum of the rows and columns in A3 is 2nk + 1. Lemma 5.1. For n = 2 (mod 4) and n ^ 34 there exists a Heffter array H(n; 3). Proof. Let n = 4m + 6 and m > 7. Let AO be: AO [2i - 1, 2i] = 8m + 1 - i, AO[2i, 2i - 1] = -(8m + i), AO [2i, 2i +1] = 12m - i, AO[2i + 1, 2i] = -(4m +1 + i), AO [2m - 2 + 2i, 2m - 1 + 2i] = 5m + i, AO[2m - 1 + 2i, 2m - 2 + 2i] = -(11m + 1 - i), AO [2m - 1 + 2i, 2m + 2i] = 9m + i, AO[2m + 2i, 2m - 1 + 2i] = -(7m +1 - i), AO[i + 1, i + 1] = -(4m - 1 - i), AO [2m + i, 2m + i] = 2m - i, AO[2m, 2m] = 4m - 1, AO[1, 4m - 13] = -(12m), AO[1,1] = 4m, AO [4m - 13,1] = 4m + 1, AO [4m - 13,4m - 13] = 6m + 6. We illustrate AO when m = 7: AO(m = 7) 28 56 -84 -57 -26 83 -30 -25 55 -58 -24 82 -31 -23 54 -59 -22 81 -32 -21 53 -60 -20 80 -33 -19 52 -61 -18 79 -34 -17 51 -62 -16 78 -35 -15 50 -63 27 36 29 -77 48 1 /A i /A m, 1 /A i /A m, 1 /A i /A m- 1, 1 /A i /A m- 1, 1 /A i /A m- 6, 1 /A i /A m- 6, 1 /A i /A m- 7, 1 /A i /A m- 7, 1 /A i /A to 2, 1 /A i /A to 14, 380 Ars Math. Contemp. 17 (2019) 349-368 Observe that the array A0 is a (4m - 13) x (4m - 13) array that has 3 filled cells in each row and column. Similarly to the previous case, this array was constructed by taking the first 4m — 14 rows and columns of the integer Heffter array H(4m; 3) given in [3], then placing entry —12m in cell (1,4m — 13), entry 4m + 1 in cell (4m — 13,1) and entry 6m + 6 in cell (4m — 13,4m — 13). Thus we need only check the sum of row 4m — 13 which is (4m + 1) + (6m + 6) — (10m + 7) = 0 and the sum of column 4m — 13 which is — 12m + (6m — 6) + (6m + 6) = 0. Hence all rows and columns in the array A0 sum to zero. We apply the same mapping as in the proof of Lemma 4.1 to the rows and columns so that all non-empty cells are a subset of the diagonals D0, Di, D2, D„_2,D„-i. Let A0 be the resultant array. The support for A0 is {1, 2, .. ., 12m + 8} \ {1, 2,. .., 13, 2m, 6m — 5, 6m — 4, .. ., 6m + 5, 6m + 7, 10m — 6, 10m — 5, ..., 10m + 6,12m + 1,12m + 2,.. ., 12m + 18}. We now arrange the 57 missing symbols into a 13 x 13 array Ai and two 3 x 3 arrays A2 and A3, where A2 and A3 ares shown below and Ai is given at the end of the Appendix. A2 A3 —8 nk — 2 — (nk — 10) — (nk — 9) —9 nk nk — 1 — (nk — 11) — 10 11 nk — 3 nk — 7 nk — 6 12 nk — 5 nk — 4 nk — 8 13 The support for Ai is {1, 2, .. ., 7, 2m, 6m — 5, 6m — 4, .. ., 6m + 5, 6m + 7, 10m — 6,10m — 5,. .., 10m + 6,12m + 1,12m + 2, .. ., 12m + 6}. Finally we give A2 and A3 with support {8, 9,10,11,12,13} U {nk — 11, nk — 10,..., nk}. In the case k = 3, observe that {nk — 11, nk — 10,..., nk} = {12m + 7,12m + 8, ..., 12m + 18}. The concatenation of A0, Ai, A2 and A3 gives a H(4m+6; 3) Heffter array for m > 7. □ Theorem 5.2. There exists a Heffter array H(n; k) for all n = 2 (mod 4) and k = 3 (mod 4), where n > k ^ 3. Proof. An H(6; 3) is given in the Appendix. Otherwise let n = 4m + 6 where m > 1. When 1 < m < 5 concatenate the B(4m+6) given in the Appendix with the array C given in Case B to get an H(4m + 6; 3). Observe that when 1 < m < 4 the entries are only on diagonals D0, Di, D2, Dn-2, Dn-i as before. When n = 26 the entries are on diagonals D0, Di, D2, D24, D25 and D9. We pair up the rest of the diagonals as {D2i, D2i+i} for 5 < i < 11, and {D3, D4}, {D5, D6}, {D7, D8} to get the required Hamilton cycles. When m = 6, n = 30; an H(30; 3) is given in the Appendix. Two Hamilton cycles H and K are also given as a reference on the array. These Hamilton cycles together with H(30; 3) only have entries on diagonals D0, Di, D2, D3, D27, D28, D29, Di8 and Dii. For the rest of the diagonals, pair them up as {D4, D5}, {D6, D7}, {D8, D9}, {Di2, Di3}, N. J. Cavenagh et al.: The existence of square non-integer Heffter arrays 381 {D14,D15}, {D16,D17} and {D2i-1,D2i} for 10 < i < 13 to get the necessary Hamilton cycles. When m > 7, a H(n; 3) exists by Lemma 5.1, then the proof follows as in Lemma 4.1. □ 6 Case D: k = 1 (mod 4) and n = 1 (mod 4) In this section we construct a Heffter array H(n; k) where n = 4m + 1 and k = 4p +1, with k < n and hence m > 2. The case H(9; 5) is given in the Appendix and so henceforth we assume m > 3. We begin with k = 5 construct an n x n array for which the sum of each row and column is 2nk + 1 = 40m + 11. First we give an example H(17; 5) of our construction. (Note a Hamilton cycle H has been included for the case k > 5.) H(17; 5) 85 -27 11 50 H H 52 68 84 -28 12 35 H H 67 83 -29 13 H H 37 66 82 -30 14 39 H H 65 81 -31 15 41 H H 64 80 -32 43 H 16 H 63 79 -33 17 45 H H H H 62 78 -34 18 47 -20 H H 61 60 21 49 H H 77 59 -19 3 51 36 H H 76 58 -22 23 38 H H 75 57 -4 5 40 H H 25 74 56 -24 42 H H 73 55 -6 7 2 44 H H 72 54 -1 9 46 H 53 71 H -8 H 10 48 -26 H 70 69 Lemma 6.1. For n = 1 (mod 4), n ^ 13 and k = 5 there exists a Heffter array H(n; 5). Proof. Let n = 4m +1 for m > 3. In every case here row and column sums will equal 40m + 11. We give the general construction of A below. A [4 m — 1,4m] = —1, A[4m — 1,1] = 2, A[4m, 2] = 2m + 1, A[4m + 1, 3] = 2m + 2, A[2m — 2,4m] = 4m, A[2m — 1, 2m + 2] = 4m + 1, A[2m, 2m + 3] = 4m + 2, A[2m + 2, 2m + 3] = —(4m + 3), A[2m +1,1] = —(4m + 4), A[1, 2m] = 12m + 2, 382 Ars Math. Contemp. 17 (2019) 349-368 A [4m +1, 2m + 1] = -(6m + 2), A[4m - 3, 2m + 1] = 6m + 1, A[2m, 2m + 2] = -(8m + 2), A[2, 2m + 1] = 8m + 3, A [3,4m] = 8m + 5, A[1,4m + 1] = 12m + 4, A[4m, 2m + 2] = 12m + 5, A[4m + 1,4m + 1] = 16m + 5. A[i, i + 3] = 2m + i + 2, 1 < i ^ 2m 3, (6.1) A[i + 1,i] = 16m + 5 - i, 1 < i ^ 2m, (6.2) A[i, i] = 20m + 6 - i, 1 < i ^ 2m, (6.3) A[2m - i, 2m + 1 - i] = -(8m + 2 - i), 1 < i ^ 2m 1, (6.4) A[2m + 3 - i, 4m + 2 - i] = 12m + 5 - 2i, 1 < i ^ 2m 1, (6.5) A[2m + i, 2m + i] = 14m + 5 - i, 1 < i ^ 2m 1, (6.6) A[2m + 1 + i, 2m + i] = 18m + 6 - i, 1 < i ^ 2m, (6.7) A[4m + 2 - i, 2m - i] = 12m + 2 - 2i, 1 < i ^ 2m 1, (6.8) A[2m + 2i, 2m + 2i + 3] = 2i + 1, 1 < i ^ m - 1, (6.9) A[2m + 2i + 2, 2m + 2i + 3] = -(2i + 2), 1 < i ^ m - 1, (6.10) A [2m + 2i - 1, 2m + 2i + 2] = 4m + 2i + 3, 1 < i ^ m - 2, (6.11) A [2m + 2i + 1, 2m + 2i + 2] = -(4m + 2i + 4), 1 < i ^ m - 2. (6.12) We note that the support of A contains: • 3,4,..., 2m by (6.9) and (6.10), • 2m + 3, 2m + 4,..., 4m - 1 by (6.1), • 4m + 5, 4m + 6,..., 6m by (6.11) and (6.12), • 6m + 3, 6m + 4,..., 8m +1 by (6.4), • 8m + 4, 8m + 6, 8m + 7,..., 12m, 12m + 1,12m + 3 by (6.5) and (6.8), • 12m + 6,12m + 7,..., 16m + 4 by (6.6) and (6.2), • 16m + 6,16m + 7,..., 20m + 5 by (6.7) and (6.3). It follows that the support of A is {1, 2,..., 20m + 5} as required. To verify the sum of each row and column is 40m +11 we begin by noting that, respectively, (6.1), (6.2), (6.3), (6.4) and (6.5) give the sum for row r (where 4 < r < 2m - 3) and (6.1), (6.2), (6.3), (6.4) and (6.8) give the sum for column c (where 4 < c < 2m - 1) as: (2m + r + 2) +(16m + 6 - r) + (20m + 6 - r) - (6m + r + 2) +(8m - 1 + 2r) = 40m + 11 and (2m + c - 1) + (16m + 5 - c) + (20m + 6 - c) - (6m + 1 + c) + (8m + 2 + 2c) = 40m +11. For row 2m + r, where 3 < r < 2m - 4 (or r = 2m - 2) and column 2m + c, where 4 < c < 2m - 1, we argue as follows. N. J. Cavenagh et al.: The existence of square non-integer Heffter arrays 383 Respectively (6.6), (6.7) and (6.8) give a partial sum of 40m + 10 for row r and (6.5), (6.6) and (6.7) give a partial sum of 40m + 12 for column c: (14m + 5 - r) + (18m + 7 - r) + (8m - 2 + 2r) = 40m + 10, (6.13) (8m + 1 + 2c) + (14m + 5 - c) + (18m + 6 - c) = 40m + 12. (6.14) Next (6.9) and (6.10) imply that if r is even, row 2m + r contain the entries r + 1 and -r giving a partial sum of 1. If r is odd, (6.11) and (6.12) imply that row 2m + r contains the entries 4m + r + 4 and - (4m + r + 3) also giving a partial sum of 1. Adding this to the partial sum in (6.13) we get an overall row sum of 40m + 11. Then (6.9) and (6.10) imply that if c is odd, column 2m + c contains the entries c - 2 and -(c - 1) giving a partial sum of -1. If c is even, (6.11) and (6.12) imply that column 2m + c contains the entries 4m + c +1 and -(4m + c + 2) also giving a partial sum of -1. Adding this to the partial sum in (6.14) we get a column sum of 40m +11. The remaining rows and columns can be checked individually to complete the proof that all rows and columns sum to 40m +11. Thus we have the required H (4m +1;5). □ Next, with care, we add up to 2(m - 2) Hamilton cycles to obtain an H(4m + 1; 4p + 5) where p < m - 2. Theorem 6.2. For n = 1 (mod 4) and k = 1 (mod 4) there exists a Heffter array H(n; k), where k < n. Proof. Let n = 4m + 1. The Heffter array H(9; 5) is given in the Appendix. Otherwise, an H(n; 5) exists by Lemma 6.1. This was labeled A in the proof of that lemma. Observe that the occupied cells of A are a subset of the union of diagonals D := D„-3 u D„-2 U Dn_i U Do u Di U D4 u D2m-4 u D2m-2 U D2m-1 U D2m U D2m+2. For each m > 3, the following set of cells is a subset of D, that does not intersect A and forms a Hamilton cycle H: {(i, 2m + 1 + i), (i, 2m + 2 + i) | 1 < i < 2m - 3} U {(2m - 2, 4m - 1), (2m - 2, 4m + 1)} U {(2m - 1, 4m + 1), (2m - 1, 4m), (4m, 4m), (4m, 2m + 1)} U {(i, i - 2m +1), (i, i - 2m + 2) | 2m < i < 4m - 1} U {(4m + 1,1), (4m +1, 2m + 2)}. (See the array H(17; 5) above for an example.) Thus there exists 4m+1 -11 = 4(m-2) -2 diagonals that do not intersect A U H and so it is possible to construct 2m - 5 disjoint Hamilton cycles by pairing empty diagonals that are either distance 1 or 2 apart. For m > 6 a possible pairing of diagonals is: {D2, D3}; {D2m-5, D2 m— {D5+2i, D6+2i}, 0 < i < m - 6; {D2m+1, D2m+3}; {D2m+4+2i, D2m+5+2i}, 0 < i < m - 4. 384 Ars Math. Contemp. 17 (2019) 349-368 When m = 4 we pair the diagonals as {D2,D3}; {D9,Du}; {D12,D13} and when m = 5 we pair the diagonals as {D2,D3}; {D5,D7}; {DU,D13}; {D14,D15}; {Die,D17}. Together with H this gives a total of 2m - 4 Hamilton cycles. Thus applying Theorem 2.2 recursively, we can form a Heffter array for each k such that k = 1 (mod 4) and k < n - 4. □ 7 Case E: k = 1 (mod 4) and n = 2 (mod 4) In this section we construct a Heffter array H(n; k) where n = 4m + 2 and k = 4p + 1, where k < n. We demonstrate the following construction in the case m = 4 with an example of an H(18; 5); the cycles H and K will be needed later for the case k > 5. H (18; 5) 2 51 20 -38 -35 -40 4 53 19 -36 -42 -28 6 37 27 39 -44 -29 8 26 41 25 -46 -30 10 43 24 -48 -31 12 45 23 -50 -32 14 47 22 -52 -33 16 49 21 -54 -34 18 1 81 K H 69 -60 K 90 H H 3 80 K H 70 -61 K 89 88 H 5 79 K H 71 -62 K 72 87 H 7 78 K H K -63 -55 K 86 H 9 77 K H 64 K -56 K 85 H 11 76 65 H 75 66 -57 K 84 H 13 H K K H 67 -58 K 83 H 15 74 H K H 68 -59 K 82 73 17 Lemma 7.1. For n = 2 (mod 4), n ^ 6 there exists a Heffter array H(n; 5). Proof. Let n = 4m + 2. The case H(n; k) = H(6; 5) is given in the Appendix and so henceforth we assume m > 2. Our Heffter array will be the concatenation of an array A0 in the first 2m + 1 rows and columns and an array A1 in the last 2m +1 rows and columns. The row and column sums of A0 will be 0 and the row and column sums of A1 will be 2nk + 1. In our definition of A0, rows and columns are calculated modulo 2m + 1 rather than n. We begin by defining a (2m +1) x (2m + 1) array AO which has support {2, 4, 5, , 4m + 2} U {4m + 3, 4m + 4, , 12m + 6} N. J. Cavenagh et al.: The existence of square non-integer Heffter arrays 385 and for which each row sums to 0 and each column sums to 0 except for columns 1 and 2m + 1, which sum to -(2m + 1) and 2m + 1, respectively. Then we swap the entry -(8m + 4) in cell (2,1) with the entry -(8m + 8) in cell (2,2m + 1) and swap the entry 6m + 2 in cell (4,1) with the entry 8m + 7 in cell (4, 2m +1). The result will be an array A0 defined on row and column set {1,2,..., 2m + 1} with row and column sums equal to 0. To this end, for 1 < i < 2m + 1 let A0M = 2i; A0[2 + i, 1 + i] = -(6m + 3 + i); AO[i, i - 2] = -(8m + 4 + 2i). Now let Ao [r, c] = < AO [3 - i, 2m + 1 - i] = 4m + 2 + i; AO [2 + i,i - 2] = 8m + 3 + 2i; -(8m + 8), (r, c) = (2,1), -(8m + 4), (r, c) = (2, 2m + 1), 8m + 7, (r, c) = (4,1), 6m + 2, (r, c) = (4, 2m + 1), _ AO [r, c], otherwise. The case m = 4 is illustrated in the example H(18; 5) given above. It will be useful to note that the non-empty cells of this (2m +1) x (2m + 1) array A0 are a subset of the union of diagonals D0 := D0 U D U D2 U D3 U D4. Next, we define a (2m + 1) x (2m + 1) array Ai, on row and column set {2m + 2, 2m + 3,..., 4m + 2} with support {1, 3,4,..., 4m + 1} U {12m + 7,12m + 8,..., 16m + 8} U {kn - 4m - 1, kn - 4m,..., kn}. For the case k = 5, observe that {kn - 4m - 1, kn - 4m,..., kn} = {16m + 9,16m + 10,..., 20m + 10}. Thus when k = 5 the support of A0 U A1 is {1,2,..., 20m + 10} as required. Each row and column of A1 will sum to 2nk + 1. We first give A1 for the cases m = 2 and m = 3 separately; then we present the general formula. A1(m = 2,n = 10) 1 10k - 4 -31 10k - 2 37 39 3 10k -34 10k - 7 -33 40 5 10k - 8 10k - 3 10k - 1 10k - 6 36 7 -35 10k - 5 -32 10k - 9 38 9 A1 (m = 3, n = 14) 1 14k -43 14k - 7 50 14k - 1 3 51 14k - 8 -44 -45 5 14k - 2 55 14k - 12 14k - 3 7 14k - 11 54 -46 56 -47 9 14k - 13 14k - 4 14k - 6 53 -48 14k - 9 11 -49 14k - 10 14k - 5 52 13 386 Ars Math. Contemp. 17 (2019) 349-368 Otherwise m > 4 and we define A1 as follows. The rows and columns are defined modulo 2m + 1 rather than modulo n. To construct the overall Heffter array, the array Ai is then shifted by adding 2m + 1 (as an integer) to each row and column. A1[4,1] = 14m + 8; A1[5, 2m + 1] = 14m + 9; (7.1) A1[6, 2m] = 16m + 8; A1[2m - 1,1] = kn - 4m + 1; (7.2) Ai[2m, 2m + 1] = kn - 4m; A1[2m + 1, 2m] = kn - 4m - 1; (7.3) A1[i,i] = 2i - 1, 1 < i < 2m + 1; (7.4) A1[i, 2m - 1 + i] = kn + 1 - i, 1 < i < 2m + 1; (7.5) A1[i, i + 1] = kn - 2m - i, 1 < i < 2m - 2; (7.6) A1[4 + i,i] = -(12m + 6 + i), 1 < i < 2m + 1; (7.7) A1 [6 + i, 1 + i] = 14m + 9 + i, 1 < i < 2m - 2. (7.8) We note that the support for A1 is the union of the sets: • {1, 3, 5, 6,..., 4m + 1} (by (7.4)), • {12m + 7,12m + 8,..., 14m + 7} (by (7.7)), • {14m + 8,14m + 9} (by (7.1)), • {14m + 10,..., 16m + 7} (by (7.8)), • {16m + 8} (by (7.2)), • {kn - 4m - 1, kn - 4m, kn - 4m + 1} (by (7.2), (7.3)), • {kn - 4m + 2,..., kn - 2m - 1} (by (7.6)) and • {kn - 2m,..., kn} (by (7.5)). We next check the row and column sums. For row r in the range 7 to 2m + 1, (7.5) and (7.6) give a partial sum of (kn +1 - r) + (kn - 2m - r) = 2kn - 2m + 1 - 2r, while (7.7) and (7.8) give a partial sum of (-12m - 6 - (r - 4)) + (14m + 9 + (r - 6)) = 2m + 1. Now combined with (7.4) the sum of these rows is (2nk - 2m +1 - 2r) + (2m + 1) + (2r - 1) = 2nk + 1, as required. For column c in the range 2 to 2m - 1, (7.5) and (7.6) give a partial sum of (kn +1 - (c + 2)) + (kn - 2m - (c - 1)) = 2kn - 2m - 2c, while (7.7) and (7.8) give a partial sum of (-12m - 6 - c) + (14m + 9 + (c - 1)) = 2m + 2. N. J. Cavenagh et al.: The existence of square non-integer Heffter arrays 387 Now combined with (7.4) the sum of these columns is (2nk - 2m - 2c) + (2m + 2) + (2c - 1) = 2nk + 1. The sum of the remaining rows and columns can be calculated individually and overall the rows and columns of A1 sum to 2nk +1 as required. Thus the concatenation of A0 with A1 gives an H(4m + 2; 5). □ Theorem 7.2. For n = 2 (mod 4), n ^ 6 and k = 1 (mod 4) there exists a Heffter array H(n; k), where k < n. Proof. Let n = 4m + 2 and k = 4p +1. A H(n; 5) exists by Lemma 7.1. Otherwise, k > 9 and m > 2. We take the array A = A0 U A1 from the previous lemma. We will construct m — 2 cycles of length n (that is, on 2(2m + 1) = n cells) in the upper left-hand (A0) and lower right-hand (A1) quadrants, and m further cycles of length n in each of the remaining quadrants. Together these form 2m - 2 disjoint 2-factors. From the proof of Lemma 7.1, within A0 there are 2m +1 - 5 = 2(m - 2) empty diagonals, which we take in pairs to obtain m - 2 cycles of length n. Next take the intersection of the last 2m + 1 rows and columns, this is the (2m +1) x (2m +1) subarray that contains A1. We will refer to diagonals within that subarray only, recalling that the rows and columns are calculated modulo 2m + 1. We aim to find m - 2 cycles of length n from this subarray. The case m = 2 is trivial and for the case m = 3, observe that the empty cells of A1 in the previous lemma form a cycle of length 14. Otherwise m > 4 and the array A1 occupies diagonals D0, D1, D2, D3, D4, D5, D7, D2m—2 and D2m. Next take the following cells H = ({(i + 1, i), (2m - 2 + i, i) | 1 < i < 2m + 1} \ {(2m - 1,1), (2m + 1, 2m)}) U {(2m - 1, 2m), (2m + 1,1)} K = ({(3 + i, i), (7 + i, i) | 1 < i < 2m + 1} \ {(4,1), (6, 2m)}) U {(4, 2m), (6,1)}. In the example H(18; 5) above these cycles are shown in cells marked by H and K, respectively. Observe that H and K form two cycles of length 4m + 2 disjoint from A1 but are a subset of D1 U D2m-2 U D2m U D3 U D7 U D5. Thus there exists 2m +1 - 9 = 2(m - 4) diagonals that do not intersect A1 U H U K. For m > 6, we can thus form m - 4 cycles of length n by taking pairs of diagonals: {De ,D8}; {D2 m — 3, D2 m— 1}; {Dg+2i, D10+2i}, 0 < i < m - 7, and when m = 5, 2m +1 = 11 so we get one cycle of length n by taking the diagonals De and D9. 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Cavenagh et al.: The existence of square non-integer Heffter arrays 389 Appendix Case B H(7; 3) 15 -13 -2 -11 14 -3 -4 -8 12 -1 10 -9 5 21 17 18 6 19 20 16 7 B(11) -1 18 -17 24 -2 -22 -23 -3 26 -16 20 -4 19 -8 -11 -9 21 -12 -10 25 -15 -13 -14 27 B(15) 1 -36 35 -34 -3 37 33 -22 -11 39 -21 -18 -14 -12 26 -15 -17 32 28 -19 -9 30 -10 -20 -16 24 -8 -13 38 -25 -29 31 -2 -4 -23 27 390 Ars Math. Contemp. 17 (2019) 349-368 CaseC H(6; 3) -1 -16 17 -11 -4 15 12 -9 -3 -2 10 -8 13 -7 -6 18 14 5 B(10) 1 22 -23 17 2 -19 -18 15 3 -24 14 10 9 11 -20 4 -16 12 13 -21 8 B(14) -34 -1 35 -2 24 -22 36 -32 -4 -23 -3 26 -20 28 -8 30 -9 -21 -10 -19 29 -11 27 -16 25 -12 -13 -14 -17 31 -15 33 -18 B (18) 1 21 -22 -36 -4 40 35 -12 -23 -17 33 -16 -11 42 -31 -28 41 -13 -18 43 -25 -26 45 -19 -14 34 -20 -30 -2 32 27 -24 -3 -15 44 -29 -8 46 -38 -39 -9 48 47 -37 -10 N. J. Cavenagh et al.: The existence of square non-integer Heffter arrays 391 o lO 1 o CO o 1—1 1 oo lO OS 1 Oi 1 oo 1 1—1 lO 1 Oi m 1 b- lO oo i—i 1 lO lO CM i—l 1 oo 1 1—1 1—1 1 lO 1 CO lO CO oo CM 1 oo i—i 1 i—i 1 b- CM 1 i—i b- CM 1 oo CM 1 boo 1 lO i—l 1 CM lO o CM 1 OS oo OS i—l 1 b- i—l 1 CM to CM 1 i—l CM 1 oo to CM oo 1 oo co 1 lO CO i—i 1 CO CM 1 oo CM CM 1 co co 1 i—i oo 1 iO co CM 1 oo oo 1 CO co 1 1 O i—i o co 1 OS CM 392 Ars Math. Contemp. 17 (2019) 349-368 Iß CO Iß 1 i—1 i—1 iß iß œ 1—1 1 CO oo 1 boo 1 CM i—1 iß CM b- lß 1 oo i—i iß oo 1 CO CM 1 iß oo 1 CO 1 oo oo 1 i—1 CM oo iß 1 iß i—i oo o CO oo oo 1 b- CM 1 CM 1 oo CM i—i CM 1 CM oo 1 CM b- O 1 i—i CO O oo 1 i—1 oo 1 œ CM 1 CM 1 i—i b- CM CO oo 1 oo CM 1 i—i 1 O 1—1 1 i—1 LO O CM 1 œ Iß oo 1 œ CO CM iß 1 i> 1—1 1 o b- CM CM 1 oo 1 o Iß 1 oo CO oo i—i 1 œ 1 oo Iß 1 oo CO CO 1—1 1 b- 1 i—i Iß CO CO CO 1 CO 1 oo 1 b- CO N. J. Cavenagh et al.: The existence of square non-integer Heffter arrays 393 1 s -a cs 1 s -a s -a CD lO 1 s -a ^ lO CO 1 S -a 1 S -a X t- to lO 1 CO 1 CD CD 1 co CO 7 o CO 1 cs 1 CD X 00 to lO 1 CO lO 1 lO to 1 ^ o ^ lO lO X lO ^ CD 1 cs 1 ■sf 1 ^ ^ co to t-7 ^ cs lO 1 CD 1 00 to lO X to ^ 00 1 ^ 00 cs 1 cs t-1 co ^ co co X o t- 7 lO 1 X cs t-cs 1 lO (N co t- CD CO 1 CO 1 iu