Random 2-cell embeddings of multistars

Jesse Campion Loth, Kevin Halasz, Tomáš Masařík, Bojan Mohar, Robert Šámal

By using permutation representations of maps, one obtains a bijection between all maps whose underlying graph is isomorphic to a graph $G$ and products of permutations of given cycle types. By using statistics on cycle distributions in products of permutations, one can derive information on the set of all $2$-cell embeddings of $G$. In this paper, we study multistars -- loopless multigraphs in which there is a vertex incident with all the edges. The well known genus distribution of the two-vertex multistar, also known as a dipole, can be used to determine the expected genus of the dipole. We then use a result of Stanley to show that, in general, the expected genus of every multistar with $n$ nonleaf edges lies in an interval of length $2/(n+1)$ centered at the expected genus of an $n$-edge dipole. As an application, we show that the face distribution of the multistar is the same as the face distribution gained when adding a new vertex to a $2$-cell embedded graph, and use this to obtain a general upper bound for the expected number of faces in random embeddings of graphs.

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