2022/05/20 by Mella, Lorenzo, Pasotti, Anita
#05B20 #05C38 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.10164
Square relative non-zero sum Heffter arrays, denoted by NHt(n;k), have been introduced as a variant of the classical concept of Heffter array. An NHt(n; k) is an n× n partially filled array with elements in ℤv, where v=2nk+t, whose rows and whose columns contain k filled cells, such that the sum of the elements in every row and column is different from 0 (modulo v) and, for every x∈ ℤv not belonging to the subgroup of order t, either x or -x appears in the array. In this paper we give direct constructions of square non-zero sum Heffter arrays with no empty cells, NHt(n;n), for every n odd, when t is a divisor of n and when t∈\2,2n,n2,2n2\. The constructed arrays have also the very restrictive property of being "globally simple"; this allows us to get new orthogonal path decompositions and new biembeddings of complete multipartite graphs.