Random 2-cell embeddings of multistars

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