Random 2-cell embeddings of multistars

TL;DR

This paper extends the analysis of random 2-cell embeddings from dipoles to multistars, providing bounds on the expected number of faces and insights into the average genus of such embeddings.

Contribution

It generalizes the expected face count results from dipoles to multistars and derives bounds for any graph based on vertex degrees.

Findings

Expected faces of multistars lie in a narrow interval around dipole expectations.

Bounds on expected faces are expressed in terms of vertex degrees.

Conjecture that expected faces grow linearly with the number of edges.

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 $G$.