# Relative Non-Zero Sum Heffter Arrays and Orthogonal Path Decompositions

I would like to present a variant introduced in [2] of the notion of an *Heffter array*.

Given a positive integer $\upsilon = 2nk +t$ with $t$ a divisor of $2nk$, a *non-zero sum Heffter array over* $\mathbb{Z}_{\upsilon}$ $relative$ $to$ $_{\frac{\upsilon}{t}}\mathbb{Z}_{\upsilon}$ (the subgroup of $\mathbb{Z}_{\upsilon}$ of order $t$) is an $n \times n$ partially filled array with entries in $\mathbb{Z}_{\upsilon}$ satisfying the following conditions: 1) each row and each column contains exactly $k$ filled cells; 2) for every $x \in \mathbb{Z}_{\upsilon} \backslash _{\frac{\upsilon}{t}}\mathbb{Z}_{\upsilon}$, either $x$ or $-x$ appears in the array; 3) the sum of the elements in every row and column is different from 0 (in $\mathbb{Z}_{\upsilon}$).

I will give some existence results showing their impact on cyclic orthogonal path decompositions of the complete multipartite graph in [2] and [3].

[1] D.S. Archdeacon, *Heffter arrays and biembedding graphs on surfaces*, Electron. J. Combin., 22 #P1.74, 2015.

[2] S. Costa, S. Della Fiore, A. Pasotti, *Non-zero sum Heffter arrays and their applications*, to appear on Discrete Math., preprint available at https://arxiv.org/abs/2109.09365.

[3] L. Mella, A. Pasotti, *Globally simple relative non-zero sum Heffter arrays and biembeddings*, in preparation.

*Bio*: Anita Pasotti received her Ph.D. in Mathematics in 2006 at the University of Milano-Bicocca, Italy. In 2010, she became an Assistant Professor at the University of Brescia where she is currently an Associate Professor of Geometry. Her area of research is Combinatorial Designs, in particular, Graph Decompositions and Heffter Arrays. She received the "Hall Medal" from the Institute of Combinatorics and its Applications in 2021 and she just became a council member of this Institute.