Partial duality for ribbon graphs, I: Distributions

The partial dual G^A with respect to a subset A of edges of a ribbon graph G was introduced by Chmutov in connection with the Jones–Kauffman and Bollobás–Riordan polynomials, and it has developed into a topic of independent interest. This paper studies, for a given G, the enumeration of the partial duals of G by Euler genus, as represented by its generating function, which we call the partial-dual Euler-genus polynomial of G. A recursion is given for subdivision of an edge and is used to derive closed formulas for the partial-dual genus polynomials of four families of ribbon graphs. The log-concavity of these polynomials is studied in some detail. We include a concise, self-contained proof that χ(G^A)=χ(A)+χ(E(G)−A)−2|V(G)|, where χ(G)=|V(G)|−|E(G)|+|F(G)|, and where A represents the ribbon graph obtained from G by deleting all edges not in A. This formula is a variant of a result of Moffatt.