  Gross, Mansour and Tucker introduced the partial-dual orientable genus
polynomial and the partial-dual Euler genus polynomial. They computed these two
partial-dual genus polynomials of four families of ribbon graphs, posed some
research problems and made some conjectures. In this paper, we introduce the
notion of signed sequences of bouquets and obtain the partial-dual Euler genus
polynomials for all ribbon graphs with the number of edges less than 4 and the
partial-dual orientable genus polynomials for all orientable ribbon graphs with
the number of edges less than 5 in terms of signed sequences. We check all the
conjectures and find a counterexample to the Conjecture 3.1 in their paper:
There is no orientable ribbon graph having a non-constant partial-dual genus
polynomial with only one non-zero coefficient. Motivated by this
counterexample, we further find an infinite family of counterexamples to the
conjecture. Moreover, we find a counterexample to the Conjecture 5.3 in their
paper: The partial-dual Euler-genus polynomial for any non-orientable ribbon
graph is interpolating.
