Partial-dual Euler-genus distributions for bouquets with small Euler genus - Yang, Yan (Ars mathematica contemporanea, 2022)
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. J. L. Gross et al. Eur. J. Comb. 86, Article ID 103084, 20 p. (2020) 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.
Keywords: Euler-genus polynomial; orientable genus polynomial; parcialni dual; partial dual; polinom Eulerjevega rodu; polinom orientabilnega rodu; ribbon graph; trakovni graf