Globally simple Heffter arrays $H(n;k)$ when $k\equiv 0,3\mod 4$

Abstract

Square Heffter arrays are $n\times n$ arrays such that each row and each 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 $1\leq x\leq nk$. Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face $2$-colourable embedding of the complete graph $K_{2nk+1}$ on an orientable surface, where for each colour, the faces give a $k$-cycle system. Moreover, a cyclic permutation on the vertices acts as an automorphism of the embedding. These necessary conditions pertain to cyclic orderings of the entries in each row and each column of the Heffter array and are: (1) for each row and each column the sequential partial sums determined by the cyclic ordering must be distinct modulo $2nk+1$; (2) the composition of the cyclic orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. We construct Heffter arrays that satisfy condition (1) whenever (a) $k\equiv 0\mod 4$; or (b) $n\equiv 1\mod 4$ and $k\equiv 3\mod 4$; or (c) $n\equiv 0\mod 4$, $k\equiv3\mod 4$ and $n\gg k$. As corollaries to the above we obtain pairs of orthogonal $k$-cycle decompositions of $K_{2nk+1}$.