论文标题
大主教类型的生物床：其完整的自动形态组及其人数
Biembeddings of Archdeacon type: their full automorphism group and their number
论文作者
论文摘要
大主教在他的开创性论文$ [1] $中定义了heffter数组的概念，以便提供$ \ mathbb {z} _ {v} $的明确构造 - 完整图的常规biembeddings $ k_v $ $ k_v $的常规biembeddings in Forcorable cormfaces。 在本文中，我们首先介绍了heffer阵列概念的概括，我们表明，在这种情况下，我们可以定义$ 2 $ - 可加油的ArchdeAcon类型的完整多Artiptite图形$ K _ {\ frac {\ frac {v} {v} {t} {t} {t} {t Times time t} $ to surece surece surece to to an noce surece to noce surece to noce surece to noce surece。然后，我们的主要目标是研究这些嵌入的完整自动形态组：在这里，我们能够使用一种概率方法证明，几乎总是，该组正好是$ \ Mathbb {z} _ {v} $。 作为此结果的应用，给定一个积极的整数$ t \ not 2 \ equiv 0 \ pmod {4} $，我们证明，对于无限的多对$ v $和$ k $，至少$（1-o（1-o（1）） $ k _ {\ frac {v} {t} \ times t} $的脸部长度为$ k $的倍数。这里$ ϕ（\ cdot）$表示Euler的基本函数。此外，如果$ t = 1 $和$ v $是素数，几乎所有这些嵌入式都定义了所有相同长度$ kV $的面孔，即，我们的非同质$ kv $ kv $ gonal biembeddings的指数数量超过$ k_ {v} $。
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 $K_{v}$.
