Biembeddings of cycle systems using integer Heffter arrays

Abstract

In this paper, we use constructions of Heffter arrays to verify the existence of face 2-colorable embeddings of cycle decompositions of the complete graph. Specifically, for n 1 (mod 4) ≡ and k ≡ n k 3(mod 4), 7 ≫ ⩾

Relative Heffter arrays and biembeddings

“…In [3], Archdeacon introduced Heffter arrays also because they are useful for finding biembeddings of cycle decompositions, as shown, for instance, in [13,17,18,20]. In this section, generalizing some of his results we show how starting from a λ-fold relative Heffter array it is possible to obtain suitable biembeddings.…”

“…The face boundaries of the embedding corresponding to ρ are the orbits of ρ • τ .Given a λ-fold relative Heffter array λ H t (m, n; s, k), say A, then the orderings ω r (associated to the permutation α r ) and ω c (associated to the permutation α c ) are said to be compatible if α c • α r is a cycle of length |skel(A)|. With the same proof of Theorem 1.4 of[13] we obtain the following necessary conditions for the existence of compatible orderings. If there exist compatible orderings ω r and ω c for a λ-fold relative Heffter array λ H t (m, n; s, k) then one of the following has to be satisfied:(1) m, n, s and k are odd;(2) m is odd, while n and s are even;(3) n is odd, while m and k are even.Theorem 5.5.…”

On the number of non-isomorphic (simple) $k$-gonal biembeddings of complete multipartite graphs

“…The focus of this paper is not the existence problem of Heffter arrays, but their connection with face 2-colorable embeddings. We point out that there are several papers in which Heffter arrays have been investigated to obtain biembeddings see [1,4,6,9,10,11,12]. To present such a connection, now we have to introduce the concepts of simple and compatible orderings.…”

“…We consider now the family of embeddings of K v obtained by Cavenagh, Donovan, and Yazıcı in [6]. In their constructions, all the face boundaries are cycles of length k.…”

“…Proposition 3.4 will provide an upper bound on this number. In the last two sections we will consider, and we will hugely extend, two families of Z vregular embeddings obtained via Heffter arrays, respectively, in [4,6] and in [9,10]. These families, together with Proposition 3.4, allow us to achieve the existence of an exponential number of non-isomorphic k-gonal biembeddings of K v and K v t ×t in several situations.…”