Partial-dual Euler-genus distributions for bouquets with small Euler genus

Abstract

For an arbitrary ribbon graph G, the partial-dual Euler-genus polynomial of G is a generating function that enumerates partial duals of G by Euler genus. When G is an orientable ribbon graph, the partial-dual orientable genus polynomial of G is a generating function that enumerates partial duals of G by orientable genus. Gross, Mansour, and Tucker inaugurated these partial-dual Euler-genus and orientable genus distribution problems in 2020. A bouquet is a one-vertex ribbon graph. Given a ribbon graph G, its partial-dual Euler-genus polynomial is the same as that of some bouquet; this motivates our focus on bouquets. We obtain the partial-dual Euler-genus polynomials for all the bouquets with Euler genus at most two.