Random 2-cell embeddings of multistars

Abstract

Random 2-cell embeddings of a given graphGGare obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces,E[FG]\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 withnnnonleaf edges lies in an interval of length2/(n+1)2/(n+1)centered at the expected number of faces of annn-edge dipole. This allows us to derive bounds onE[FG]\mathbb {E}[F_G]for any given graphGGin terms of vertex degrees. We conjecture thatE[FG]≤O(n)\mathbb {E}[F_G]\le O(n)for any simplenn-vertex graphGG.