论文标题
反示例概述了毛，曼苏尔和塔克在色带图的部分偶属多项式上
Counterexamples to conjectures by Gross, Mansour and Tucker on partial-dual genus polynomials of ribbon graphs
论文作者
论文摘要
Gross，Mansour和Tucker引入了部分双重定位属多项式和部分双重欧拉属多项式。他们计算了四个色带图家族的这两个部分双属多项式属，提出了一些研究问题并做出了一些猜想。在本文中，我们介绍了花束的签名序列的概念，并为所有功能区图获得了部分双重欧拉属多项式，其边缘数小于4，并且对于所有可定向的色带图的偏边数量均小于4，而具有符号序列的5个偏差的多项式属。我们检查了所有猜想，并在论文中找到了猜想3.1的反例：没有可定向的色带图具有非恒定部分二线属多项式，只有一个非零系数。在这个反例的激励下，我们进一步发现了一个无限的反示例家族。此外，我们在论文中找到了对猜想5.3的反例：对于任何不可方向的色带图，部分双二重奏欧拉 - 基因多项式都在插值。
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.
