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Random 2-cell embeddings of multistars

Jesse Campion Loth, Kevin Halasz, Tomáš Masařík, Bojan Mohar, Robert Šámal

TL;DR

This work analyzes the average number of faces $\mathbb{E}[F_G]$ in uniform random orientable embeddings of a graph $G$, equating to the average genus. It uses local-rotation representations and Stanley's generating functions to obtain an exact expression for the dipole case and a tight, near-dipole bound for multistars, depending only on the partition type and a simple harmonic-mean quantity $\Delta_{n'}$. These results are extended to general graphs via a vertex-adding framework that yields practical upper bounds such as $\mathbb{E}[F] = O(n)$ in many families and a bound $\mathbb{E}[F] < n(\log d+5/3)$ for graphs of maximum degree $d$, along with $d$-degenerate bounds. The paper also posits a conjecture that $\mathbb{E}[F_G] = O(n)$ for all simple graphs, supported by linear-growth examples and degree-based analyses. Overall, it provides precise tools to bound topological complexity of random graph embeddings and connects dipole/multistar structures to broader graph classes.

Abstract

Random 2-cell embeddings of a given graph $G$ are obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces, $\mathbb{E}[F_G]$, of such an embedding which is equivalent to studying its average genus. So far, tight results are known for two families called monopoles and dipoles. We extend the dipole result to a more general family called multistars, i.e., loopless multigraphs in which there is a vertex incident with all the edges. In particular, we show that the expected number of faces of every multistar with $n$ nonleaf edges lies in an interval of length $2/(n + 1)$ centered at the expected number of faces of an $n$-edge dipole. This allows us to derive bounds on $\mathbb{E}[F_G]$ for any given graph $G$ in terms of vertex degrees. We conjecture that $\mathbb{E}[F_G ] \le O(n)$ for any simple $n$-vertex graph $G$.

Random 2-cell embeddings of multistars

TL;DR

This work analyzes the average number of faces in uniform random orientable embeddings of a graph , equating to the average genus. It uses local-rotation representations and Stanley's generating functions to obtain an exact expression for the dipole case and a tight, near-dipole bound for multistars, depending only on the partition type and a simple harmonic-mean quantity . These results are extended to general graphs via a vertex-adding framework that yields practical upper bounds such as in many families and a bound for graphs of maximum degree , along with -degenerate bounds. The paper also posits a conjecture that for all simple graphs, supported by linear-growth examples and degree-based analyses. Overall, it provides precise tools to bound topological complexity of random graph embeddings and connects dipole/multistar structures to broader graph classes.

Abstract

Random 2-cell embeddings of a given graph are obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces, , of such an embedding which is equivalent to studying its average genus. So far, tight results are known for two families called monopoles and dipoles. We extend the dipole result to a more general family called multistars, i.e., loopless multigraphs in which there is a vertex incident with all the edges. In particular, we show that the expected number of faces of every multistar with nonleaf edges lies in an interval of length centered at the expected number of faces of an -edge dipole. This allows us to derive bounds on for any given graph in terms of vertex degrees. We conjecture that for any simple -vertex graph .

Paper Structure

This paper contains 5 sections, 10 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our results
  3. The dipole
  4. Multistars
  5. General graphs

Key Result

Theorem 3

The number of cyclic permutations $\tau \in C_n$ such that the product $(1 \, 2 \, 3 \, \dots \, n) \tau$ has $k$ cycles is equal to $\tfrac{2}{n(n+1)} c(n+1,k)$ if $n-k$ is even (and is zero if $n-k$ is odd).

Figures (2)

  • Figure 1: An embedding of $K_4$ with two faces, represented in two different ways: by using local rotations and by exposing the facial walks.
  • Figure 2: An example of the process of adding a vertex $v$ to an embedding of a graph $G$.

Theorems & Definitions (19)

  • Conjecture 1
  • Conjecture 2
  • Theorem 3: Stanley St11
  • Corollary 4
  • proof
  • Theorem 5: St11
  • Theorem 6
  • proof : Proof of Theorem \ref{['thm:asymmulti']}
  • Definition 7
  • Lemma 8
  • ...and 9 more