Gross，Mansour和Tucker关于带状图的部分对偶亏格多项式的证明的反例
Counterexamples to conjectures by Gross, Mansour and Tucker on partial-dual genus polynomials of ribbon graphs
摘要：
Gross，Mansour和Tucker引入了部分对偶可定向亏格多项式和部分对偶Euler亏格多项式。他们计算了四类带状图的这两个部分对偶亏格多项式，提出了一些研究问题，并作了一些说明。本文引入了带号丛序列的概念，利用带号序列得到了边数小于4的带图的部分对偶Euler亏格多项式和边数小于5的可定向带图的部分对偶可定向亏格多项式.我们检查了所有的猜想，并找到了他们的论文中的猜想3.1的反例：没有定向带状图具有一个非常数的部分对偶亏格多项式，只有一个非零系数。受这个反例的启发，我们进一步找到了这个猜想的无穷多个反例。此外，我们还发现了他们的猜想5.3的一个反例：任何不可定向带状图的部分对偶欧拉亏格多项式是内插的。
英文摘要：
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.
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