Non-zero sum Heffter arrays and their applications

Biembeddings of Archdeacon type: their full automorphism group and their number

“…Reasoning as in [10] and in [13] (see also Proposition 5.6 of [27]), we can prove that, since the orderings ω r and ω c are compatible, the map ρ is a rotation of K 2nk+t t ×t and we get the following Theorem.…”

“…We denote by λ c the minimum positive integer such that λc|E(C)|−1 i=0 ω i c (a) = 0 in Z 2nk+t . Due to Theorem 6.4 of [10] (see also [27]) (x, x + a) belongs to the face F 1 whose boundary is…”

“…We denote by E1 the event that the array A is an NH t (m, n; h, k). Due to the proof of Theorem 4.2 of [10] we have that, if we choose uniformly at random an element of A, then the expected value of the variable X defined by the number of rows and columns that sum to zero is…”

“…Archdeacon, in his seminal paper [1], showed that, if some additional technical conditions are satisfied, Heffter arrays provide explicit constructions of Z v -regular biembeddings of complete graphs K v into orientable surfaces. Following [10,14] and [15] the embeddings defined, using this construction, via partially filled arrays, will be denoted as embeddings of Archdeacon type or, more simply, Archdeacon embeddings. Indeed, this kind of embedding can be considered also for more general arrays than the Heffter's.…”

“…We also refer to [30], and to the references therein, for the known results on the existence problem of relative Heffter arrays. More recently, in [10] the authors introduced a variation of the Heffter arrays, denoted by non-zero sum Heffter arrays and showed that, also in this case, that embedding is well defined. In this paper, we first provide a generalization of both the (relative) Heffter and the non-zero sum Heffter arrays, and then we define the Archdeacon embedding in this more general context.…”