A class of highly symmetric Archdeacon embeddings

Abstract

Archdeacon, in his seminal paper $[1]$, defined the concept of Heffter array to provide explicit constructions of biembeddings of the complete graph $K_v$ into orientable surfaces, the so-called Archdeacon embeddings, and proved that these embeddings are $\mathbb{Z}_{v}$-regular. In this paper, we show that an Archdeacon embedding may admit an automorphism group that is strictly larger than $\mathbb{Z}_{v}$. Indeed, as an application of the interesting class of arrays recently introduced by Buratti in $[2]$, we exhibit, for infinitely many values of $v$, an embedding of this type having full automorphism group of size ${v \choose 2}$ that is the largest possible one.