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

Simone Costa

TL;DR

The work extends Archdeacon-type biembeddings by introducing quasi-Heffter and non-zero sum variants (denoted $ ext{QH}_t$ and $ ext{NH}_t$) and shows how these arrays induce $\mathbb{Z}_v$-regular, two-color embeddability of $K_{\frac{2nk+t}{t}\times t}$. A central result is that, under a probabilistic model, the full automorphism group of these embeddings is almost surely the cyclic group $\mathbb{Z}_v$, implying that highly symmetric automorphism groups are rare in this construction. The authors then translate these automorphism results into counting statements, proving infinite families with many nonisomorphic embeddings whose face lengths are multiples of $k$, with stronger abundance when $t=1$ and $v$ is prime, where most faces become of length $kv$. They further connect the combinatorics to the Crazy Knight's Tour Problem, showing solvability yields Archdeacon embeddings, and they provide rigorous lower bounds on the number of nonisomorphic Archdeacon embeddings, highlighting a large, dense landscape of such maps beyond the highly symmetric cases.

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})!}{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 $K_{v}$.

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

TL;DR

The work extends Archdeacon-type biembeddings by introducing quasi-Heffter and non-zero sum variants (denoted and ) and shows how these arrays induce -regular, two-color embeddability of . A central result is that, under a probabilistic model, the full automorphism group of these embeddings is almost surely the cyclic group , implying that highly symmetric automorphism groups are rare in this construction. The authors then translate these automorphism results into counting statements, proving infinite families with many nonisomorphic embeddings whose face lengths are multiples of , with stronger abundance when and is prime, where most faces become of length . They further connect the combinatorics to the Crazy Knight's Tour Problem, showing solvability yields Archdeacon embeddings, and they provide rigorous lower bounds on the number of nonisomorphic Archdeacon embeddings, highlighting a large, dense landscape of such maps beyond the highly symmetric cases.

Abstract

Archdeacon, in his seminal paper , defined the concept of Heffter array in order to provide explicit constructions of -regular biembeddings of complete graphs 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 -colorable embedding of Archdeacon type of the complete multipartite graph 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 . As an application of this result, given a positive integer , we prove that there are, for infinitely many pairs of and , at least non-isomorphic biembeddings of whose face lengths are multiples of . Here denotes the Euler's totient function. Moreover, in case and is a prime, almost all these embeddings define faces that are all of the same length , i.e. we have a more than exponential number of non-isomorphic -gonal biembeddings of .
Paper Structure (5 sections, 27 theorems, 106 equations, 1 figure)

This paper contains 5 sections, 27 theorems, 106 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Archdeacon Embedding
  3. On the Automorphism group
  4. Crazy Knight's Tour Problem
  5. Non-isomorphic embeddings

Key Result

Theorem 2.8

Let $A$ be a $\mathrm{Q}\mathrm{H}_t(m,n;h,k)$ that admits two compatible orderings $\omega_r$ and $\omega_c$. Then there exists a cellular biembedding $\psi$ of $K_{\frac{2nk+t}{t}\times t}$, such that every edge is on a face whose boundary length is a multiple of $h$ and on a face whose boundary l

Figures (1)

  • Figure 1: A biembedding of $K_7$ into the torus. This picture is taken from JS.

Theorems & Definitions (60)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 1.4
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5: Archdeacon embedding
  • Example 2.6
  • ...and 50 more