Biembeddings of Archdeacon type: their full automorphism group and their number
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2022
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914170986823680 |
|---|---|
| author | Costa, Simone |
| author_facet | Costa, Simone |
| contents | 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})!}{ϕ(v)} $ non-isomorphic biembeddings of $K_{\frac{v}{t}\times t}$ whose face lengths are multiples of $k$. Here $ϕ(\cdot)$ 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 $K_{v}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_02066 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Biembeddings of Archdeacon type: their full automorphism group and their number Costa, Simone Combinatorics 05C10, 05C60, 05B20, 05C30, 05C15, 05D40 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})!}{ϕ(v)} $ non-isomorphic biembeddings of $K_{\frac{v}{t}\times t}$ whose face lengths are multiples of $k$. Here $ϕ(\cdot)$ 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 $K_{v}$. |
| title | Biembeddings of Archdeacon type: their full automorphism group and their number |
| topic | Combinatorics 05C10, 05C60, 05B20, 05C30, 05C15, 05D40 |
| url | https://arxiv.org/abs/2205.02066 |