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

TL;DR

This paper introduces quasi-Heffter arrays for biembeddings of complete multipartite graphs, studies their automorphism groups, and demonstrates the abundance of non-isomorphic embeddings with specific face length properties.

Contribution

It generalizes Heffter arrays to quasi-Heffter arrays, analyzes automorphism groups probabilistically, and proves the existence of many non-isomorphic biembeddings with prescribed face lengths.

Findings

Automorphism group is almost always cyclic of order v.

Infinitely many non-isomorphic biembeddings with face lengths multiple of k.

Exponential number of k*v-gonal biembeddings for prime v and t=1.

Abstract

Archdeacon, in his seminal paper $[1]$, defined the concept of Heffter array in order to provide explicit constructions of $\mathbb{Z}_{v}$-regular biembeddings of complete graphs $K_v$ into orientable surfaces. In this paper, we first introduce the quasi-Heffter arrays as a generalization of the concept of Heffer array and we show that, in this context, we can define a $2$-colorable embedding of Archdeacon type of the complete multipartite graph $K_{\frac{v}{t}\times t}$ into an orientable surface. Then, our main goal is to study the full automorphism groups of these embeddings: here we are able to prove, using a probabilistic approach, that, almost always, this group is exactly $\mathbb{Z}_{v}$. As an application of this result, given a positive integer $t\not\equiv 0\pmod{4}$, we prove that there are, for infinitely many pairs of $v$ and $k$, at least $(1-o(1)) \frac{(\frac{v-t}{2})!}{\phi(v)} $ non-isomorphic biembeddings of $K_{\frac{v}{t}\times t}$ whose face lengths are multiples of $k$. Here $\phi(\cdot)$ denotes the Euler's totient function. Moreover, in case $t=1$ and $v$ is a prime, almost all these embeddings define faces that are all of the same length $kv$, i.e. we have a more than exponential number of non-isomorphic $kv$-gonal biembeddings of $K_{v}$.