Expected number of faces in a random embedding of any graph is at most linear

TL;DR

This paper proves that the expected number of faces in a random embedding of any graph is at most linear in the number of vertices, confirming a conjecture and improving previous bounds.

Contribution

The paper establishes a linear upper bound on the expected number of faces in random graph embeddings, generalizing prior results and confirming a longstanding conjecture.

Findings

Expected number of faces is at most n(1+H_m) for any n-vertex multigraph.

Bound is tight up to a constant factor.

Confirms the conjecture that the expected number of faces is at most linear.

Abstract

A random 2-cell embedding of a given graph $G$ is 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. In 1991, Stahl proved that the expected number of faces in a random embedding of an arbitrary graph of order $n$ is at most $n\log(n)$. While there are many families of graphs whose expected number of faces is $\Theta(n)$, none are known where the expected number would be super-linear. This lead to the conjecture that there is a linear upper bound. In this note we confirm the conjecture by proving that for any $n$-vertex multigraph, the expected number of faces in a random 2-cell embedding is at most $n(1+H_m)$, where $m$ is the maximum edge-multiplicity and $H_m$ denotes the $m$th harmonic number. This bound is best possible up to a constant factor.