Random Embeddings of Graphs: The Expected Number of Faces in Most Graphs is Logarithmic

TL;DR

The paper investigates random 2-cell embeddings of graphs on orientable surfaces via rotation systems and proves that the expected number of faces $E[F(G)]$ is typically $\Theta(\log n)$ across several graph models. It develops two complementary proof strategies: a log-square bound (simpler, weaker) and a refined logarithmic bound (central result) using a Stahl-inspired framework with partial maps and temporary faces. For complete graphs $K_n$, the authors establish tight-ish bounds $(1/2)\ln n - 2 < E[F(K_n)] \le 3.65\ln n + o(1)$ and extend the logarithmic scaling to random graphs $G(n,p)$ as well as degree-bounded families $B(n,Δ)$, illustrating why Monte Carlo methods struggle to approximate minimum genus. The work also furnishes a general $O((\ln n)^2)$ bound for $G(n,p)$ and demonstrates $\Theta(\log n)$ behavior for random multigraphs and simple graphs with fixed degree sequences, offering a broad, unify­ing view of faces in random embeddings with potential implications for related topological and probabilistic studies.

Abstract

A random 2-cell embedding of a connected graph $G$ in some orientable surface is obtained by choosing a random local rotation around each vertex. Under this setup, the number of faces or the genus of the corresponding 2-cell embedding becomes a random variable. Random embeddings of two particular graph classes, those of a bouquet of $n$ loops and those of $n$ parallel edges connecting two vertices, have been extensively studied and are well-understood. However, little is known about more general graphs. The results of this paper explain why Monte Carlo methods cannot work for approximating the minimum genus of graphs. In his breakthrough work [Permutation-partition pairs, JCTB 1991], Stahl developed the foundation of "random topological graph theory". Most of his results have been unsurpassed until today. In our work, we analyze the expected number of faces of random embeddings (equivalently, the average genus) of a graph $G$. It was very recently shown that for any graph $G$, the expected number of faces is at most linear. We show that the actual expected number of faces $F(G)$ is almost always much smaller. In particular, we prove: 1) $\frac{1}{2}\ln n - 2 < \mathbb{E}[F(K_n)] \le 3.65 \ln n +o(1)$. 2) For random graphs $G(n,p)$ ($p=p(n)$), we have $\mathbb{E}[F(G(n,p))] \le \ln^2 n+\frac{1}{p}$. 3) For random models $B(n,Δ)$ containing only graphs, whose maximum degree is at most $Δ$, we obtain stronger bounds by showing that the expected number of faces is $Θ(\log n)$.

Figures (11)

• Figure 1: An example of \ref{['def:rpA']}, processing vertex $v_3$.
• Figure 2: The upper diagram shows the step of Random Process $A$ where we are processing dart $d_3$ at vertex $v_3$. The partial facial walk is traced in dotted red line, showing the only dart for which pairing with makes a completed face. At the next step, the only dart for which pairing with makes a completed face is at vertex $v_5$. However, we have already added the edge from $v_3$ to $v_5$, so $3$ is not a valid choice of a label at this step. Therefore, we cannot add a completed face at this step.
• Figure 3: An example of \ref{['rpb']}, processing vertex $v_3$ to obtain $R_3$. At the end of this step, the darts $d_1', d_2'$ remain unpaired. It is not decided which one will go to $v_1$ and which one will go to $v_2$.
• Figure 4: Computer evaluated bound given by \ref{['eq:logsqForComputation']} for $6\le n\le 242$ in the left. In the right chart, for $243\le n\le 558$, we provide the upper bound of $5\ln(n)+5$.
• Figure 5: Computer evaluated upper bound of $5\ln(n)$ for $559\le n \le 40748$.

Theorems & Definitions (77)

• Theorem 1.1: Stahl Stahl1995
• Conjecture 1.2: Mauk and Stahl Mauk1996
• Theorem 1.3: Stahl St91JCTB
• Theorem 1.4: Campion Loth and Mohar loth2022expected
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Proposition 1.8
• Theorem 1.9