Ordering and Reordering: Using Heffter Arrays to Biembed Complete Graphs

In this paper we extend the study of Heffter arrays and the biembedding of graphs on orientable surfaces first discussed by Archdeacon in 2014. We begin with the definitions of Heffter systems, Heffter arrays, and their relationship to orientable biembeddings through current graphs. We then focus on two specific cases. We first prove the existence of embeddings for every K_(6n+1) with every edge on a face of size 3 and a face of size n. We next present partial results for biembedding K_(10n+1) with every edge on a face of size 5 and a face of size n. Finally, we address the more general question of ordering subsets of Z_n take away {0}. We conclude with some open conjectures and further explorations

Biembeddings of cycle systems using integer Heffter arrays

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 ≡3(mod 4), n k ≫ ⩾ 7 and when n ≡ 0(mod 3) then k ≡ 7(mod 12), there exist face 2-colorable embeddings of the complete graph K₂ₙₖ₊₁ onto an orientable surface where each face is a cycle of a fixed length k. In these embeddings the vertices of K₂ₙₖ₊₁ will be labeled with the elements of Z₂ₙₖ₊₁ in such a way that the group, (Z₂ₙₖ₊₁, +) acts sharply transitively on the vertices of the embedding. This result is achieved by verifying the existence of nonequivalent Heffter arrays, H (n ; k), which satisfy the conditions: (1) for each row and each column the sequential partial sums determined by the natural ordering must be distinct modulo 2nk + 1; (2) the composition of the natural orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. The existence of Heffter arrays H (n ; k) that satisfy condition (1) was established earlier in Burrage et al. and in this current paper, we vary this construction and show, for k ⩾ 11, that there are at least (n − 2)[((k − 11)/4)!/ ]² such nonequivalent H (n ; k) that satisfy both conditions (1) and (2)