Globally Simple Heffter Arrays $H(n;k)$ with $k \equiv 1 \pmod{4}$

Abstract

Heffter arrays are combinatorial structures used to construct orthogonal cyclic cycle decompositions and biembeddings of complete graphs onto surfaces. A Heffter array $H(m,n;h,k)$ is an $m \times n$ partially filled array with distinct nonzero entries from $\mathbb{Z}_{2nk+1}$ such that each row contains $h$ filled cells, each column contains $k$ filled cells, the elements in the filled cells form a half-set of $\mathbb{Z}_{2nk+1}$, and every row and column sums to zero modulo $2nk+1$. If these row and column sums equal zero over the integers, the structure is called an integer Heffter array. Furthermore, such an array is called globally simple if the partial sums of the entries in each row and column, evaluated in their natural order, are distinct modulo $2nk+1$. When $m=n$ and $h=k$, the array is square and denoted by $H(n;k)$. While the existence of globally simple square Heffter arrays has been established for several congruence classes, the cases where $k \equiv 1,2 \pmod{4}$ for $k > 10$ have remained an open problem [1]. In this work, we address this gap in the literature by explicitly constructing globally simple integer Heffter arrays $H(n;k)$ for the previously open cases where $k \equiv 1 \pmod{4}$ and $n \equiv 0,3 \pmod{4}$. Consequently, these constructions guarantee the existence of orthogonal cyclic $k$-cycle decompositions of the complete graph $K_{2nk+1}$ for these parameters. [1] J.H. Dinitz and A. Pasotti. A survey of Heffter arrays. In C.J. Colbourn, editor, New Advances in Designs, Codes and Cryptography, volume 86, pages 353-392. Springer Nature Switzerland, 2024.