Biembeddings of cycle systems using integer Heffter arrays

TL;DR

This paper proves the existence of infinitely many face 2-colourable biembeddings of complete graphs with cycles of fixed length on orientable surfaces, using novel constructions of Heffter arrays with specific properties.

Contribution

It introduces new conditions and constructions for Heffter arrays that enable the creation of biembeddings with prescribed face lengths on surfaces.

Findings

Existence of numerous non-isomorphic biembeddings for infinitely many cycle lengths.

Construction of Heffter arrays satisfying specific sum and permutation conditions.

Quantitative lower bounds on the number of such biembeddings.

Abstract

In this paper we will show the existence of a face $2$-colourable biembedding of the complete graph onto an orientable surface where each face is a cycle of a fixed length $k$, for infinitely many values of $k$. In particular, under certain conditions, we show that there exists at least $(n-2)[(p-2)!]^2/(e^2 kn)$ non-isomorphic face $2$-colourable biembeddings of $K_{2nk+1}$ in which all faces are cycles of length $k=4p+3$. These conditions are: $n\equiv 1\mod 4$, $k\equiv 3\mod 4$ and either $n$ is prime or $n\gg k$ and $n\equiv 0\mod 3$ implies $p\equiv 1\mod 3$. To achieve this result we begin by verifying the existence of $(n-2)[(p-2)!/e]^2$ non-equivalent 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 \cite{BCDY} and in this current paper we vary this construction and show that there are at least $(n-2)[(p-2)!/e]^2$ such non-equivalent $H(n;k)$ that satisfy condition (1) and then show that each of these Heffter arrays also satisfy condition (2) under certain conditions.