# The Existence of Square Integer Heffter Arrays

An integer Heffter array H(m,n;s,t) is an m × n partially filled
matrix with entries from the set {-ms,...,-2,-1,1,2,...,ms} 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_{2ms+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 Yazici 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.