Globally simple Heffter arrays H(n;k) when k≡0

Abstract

Square Heffter arrays are n × n arrays such that each row and each 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.Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face 2-colourable embedding of the complete graph K 2nk+1 on an orientable surface, where for each colour, the faces give a k-cycle system. Moreover, a cyclic permutation on the vertices acts as an automorphism of the embedding. These necessary conditions pertain to cyclic orderings of the entries in each row and each column of the Heffter array and are: (1) for each row and each column the sequential partial sums determined by the cyclic ordering must be distinct modulo 2nk+1; (2) the composition of the cyclic orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. We construct Heffter arrays that satisfy condition (1) whenever (a) k is divisible by 4; or (b) n ≡ 1 (mod 4) and k ≡ 3 (mod 4); or (c) n ≡ 0 (mod 4), k ≡ 3 (mod 4) and n ≫ k. As corollaries to the above we obtain pairs of orthogonal k-cycle decompositions of K 2nk+1 .In 1896 Heffter, [19], introduced his now famous first difference problem: partition the set {1, . . . , 3m} into m triples {a, b, c} such that either a + b = c or a + b + c is divisible by 6m + 1. However, it was not until 1939 that Peltesohn, [22], showed that a solution exists whenever m = 3. A key interest in this problem is that solutions to Heffter's first difference problem yield cyclic Steiner triple systems; see [13]. A natural extension to this question is: can we identify a set of m subsets {x 1 , . . . , x s } ⊂ {−ms, . . . , −1, 1, . . . , ms} such that the sum of the entries in each subset is divisible by 2ms + 1 and further if x occurs in one of the subsets, −x does not occur in any of the subsets? We call the set of m such subsets a Heffter system. Two Heffter systems, H 1 = {H 11 , . . . , H 1m }, |H 1i | = s for i = 1, . . . m, and H 2 = {H 21 , . . . , H 2n } |H 2j | = r for j = 1, . . . n, where sm = nt, are said to be orthogonal if for all i, j, |H 1i ∩ H 2j | 1. As observed by Dinitz and Mattern [14], a Heffter system is equivalent to a Heffter array H(m, n; s, t), which is an m × n array of integers, such that:• each row contains s filled cells and each column contains t filled cells;• the elements in every row and column sum to 0 in Z 2ms+1 ; and• for each integer 1 x ms, either x or −x appears in the array.Henceforth the set of integers {0, 1, . . . , n − 1} is denoted by [n]. In the current paper the rows and columns of an m × n array will be indexed by [m] and [n], respectively. A Heffter array is square, if m = n and necessarily s = t, and is denoted H(n; t).The following is an example of a pair of orthogonal Heffter systems that are equivalent to a Heffter array H(6, 12; 8, 4), given by Archdeacon in [2].Example 1.1. Let m = 6 and s = 8. Then for each set

Biembeddings of cycle systems using integer Heffter arrays

“…First, we work toward a proof of Theorem 1.5. In [5], constructions were provided that verify the existence of the globally simple Heffter arrays . Thus, it suffices to show that the particular Heffter arrays have orderings which are compatible and simple.…”

“…Note that a support shifted integer Heffter array is in fact an integer Heffter array . Support shifted simple integer Heffter arrays were constructed for all and in [5]. Then these arrays for were merged with a Heffter array to obtain simple Heffter arrays .…”

“…Then these arrays for were merged with a Heffter array to obtain simple Heffter arrays . In this section we first document the existing constructions from [5] then we will generalize these constructions to obtain nonequivalent support shifted simple . Then as in [5] we will merge each of these arrays with Heffter arrays to prove Theorem 1.8.…”

“…In [5] a globally simple array was constructed as follows. Let , and gcd.…”

“…In [3,11,14] it was verified that there exists an integer , and , that admit both simple and compatible orderings, when and simple orderings when . Focusing on simple orderings only, the authors of [5] constructed simple Heffter arrays, , satisfying the conditions: (a) ; or (b) and ; or (c) , and . A new type of Heffter array, called a relative Heffter array , in which certain entries are excluded with the aim of embedding complete multipartite graphs rather than complete graphs, is studied in [12].…”