两类花束图的部分对偶欧拉亏格多项式

Partial-dual Euler-genus Polynomials for Two Classes of Bouquets

[European J.Combin.,2020,86:Paper No.103084,20 pp.]在带子图中引入了部分对偶欧拉亏格多项式的概念,并给出插值猜想,即任意不可定向带子图的部分对偶欧拉亏格多项式是插值的.[European J.Combin.,2022,102:Paper No.103493,7 pp.]给出了两类反例否定了插值猜想,这两类花束图含有的侧面环只有一条或者两条不可定向环.本文是在[European J.Combin.,2022,102:Paper No.103493,7 pp.]的基础上,进一步计算其它两类花束图的部分对偶欧拉亏格多项式,其中一类是非插值的,它的侧面环有任意条不可定向环;而另一类是插值的,它的侧面环有任意条可定向环和不可定向环.

[European J.Combin.,2020,86:Paper No.103084,20 pp.]introduced the partial-dual Euler-genus polynomial of the ribbon graphs and gave the interpolating conjecture.That is,the partial-dual Euler-genus polynomial for any non-orientable ribbon graph is interpolating.In fact,[European J.Combin.,2022,102:Paper No.103493,7 pp.]have given two classes of counterexamples to prove that the conjecture is wrong which contain only one or two non-orientable side loops.On the basis of[European J.Combin.,2022,102:Paper No.103493,7 pp.],we further calculate the partial-dual Euler-genus polynomials of two other classes of bouquets.One is non-interpolating whose all side loops are non-orientable loops and the number of them can be chosen arbitrarily.The other is interpolating whose side loops are a combination of non-orientable loops and orientable loops,and the number of both can be chosen arbitrarily.