The existence of square non-integer Heffter arrays - Cavenagh, Nicholas J. (Ars mathematica contemporanea, 2019)
A Heffter array ?$H(n; k)$? is an ?$n × n$? 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 ?$1 \leqslant x \leqslant nk$?. Heffter arrays are useful for embedding the graph ?$K_{2nk + 1}$? 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 D. S. Archdeacon et al. Des. Codes Cryptography 77, No. 2-3, 409-426 (2015) and J. H. Dinitz and I. M. Wanless Ars Math. Contemp. 13, No. 1, 81-93 (2017). In this paper we consider square Heffter arrays that are not necessarily integer. We show that such Heffter arrays exist whenever ?$3 \leqslant k < n$?.
Keywords: biembedding cycle systems; bivložitveni ciklični sistemi; Heffter arrays; Heffterjeve matrike