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 2 n k + 1 and either x or − x appears in the array for each integer 1 ⩽ x ⩽ n k . Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face 2-colourable embedding of the complete graph K 2 n k + 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 2 n k + 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 ≡ 0 ( mod 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 2 n k + 1 .

中文翻译：

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当 k≡0,3(mod4) 时，全局简单 Heffter 数组 H(n;k)

Abstract Square Heffter 数组是 n × n 数组，每行和每列包含 k 个填充单元格，每行和每列总和可被 2 nk + 1 整除，并且对于每个整数 1 ⩽ x ⩽，x 或 - x 出现在数组恩克。Archdeacon 指出，满足两个附加条件的 Heffter 阵列会在可定向表面上产生完整图 K 2 nk + 1 的面 2 色嵌入，其中对于每种颜色，面给出 ak 循环系统。此外，顶点上的循环置换充当嵌入的自同构。这些必要条件与 Heffter 数组的每一行和每一列中条目的循环排序有关，并且是： (1) 对于每一行和每列，由循环排序确定的连续部分和必须是不同的模 2 nk + 1 ；(2) 行和列的循环排序的组合相当于数组中条目的单个循环排列。当 (a) k ≡ 0 ( mod 4 ) 时，我们构造满足条件 (1) 的 Heffter 数组；或 (b) n ≡ 1 ( mod 4 ) 和 k ≡ 3 ( mod 4 ) ；或 (c) n ≡ 0 ( mod 4 ) , k ≡ 3 ( mod 4 ) 和 n ≫ k 。作为上述的推论，我们获得了 K 2 nk + 1 的正交 k 循环分解对。