Abstract
An integer Heffter array H ( m , n ; s , t ) is an m × n partially filled matrix with entries from the set { ± 1, ± 2, …, ± m s } such that i) each row contains s filled cells and each column contains t filled cells, ii) every row and column sums to 0 (in Z ), and iii) no two entries agree in absolute value. Heffter arrays are useful for embedding the complete graph K 2 m s + 1 on an orientable surface in such a way that each edge lies between a face bounded by an s -cycle and a face bounded by a t -cycle. In 2015, Archdeacon, Dinitz, Donovan and Yaz i c i constructed square (i.e. m = n ) integer Heffter arrays for many congruence classes. In this paper we construct square integer Heffter arrays for all the cases not found in that paper, completely solving the existence problem for square integer Heffter arrays.
Highlights
A Heffter array H(m, n; s, t) is an m × 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, m = n and necessarily s = t
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
Given a Heffter array H(m, n; s, t) with compatible orderings ωr of the symbols in the rows of the array and ωc on the symbols in the columns of the array, there exists an embedding of K2ms+1 such that every edge is on a face of size s and a face of size t
Summary
A Heffter array H(m, n; s, t) is an m × n matrix of integers such that:. (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. Given a Heffter array H(m, n; s, t) with compatible orderings ωr of the symbols in the rows of the array and ωc on the symbols in the columns of the array, there exists an embedding of K2ms+1 such that every edge is on a face of size s and a face of size t. 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. If A is an array with support S and z a nonnegative integer, A ± z has support S + z
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