Partial-dual Euler-genus polynomials for two classes of bouquets
Kefu ZHU1(), Qi YAN2,3()
. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China . School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China . School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
[European J. Combin., 2020, 86: Paper No. 103084, 20 pp.] introduced the concept of partial-dual Euler-genus polynomial in the ribbon graphs and gave the interpolation 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.] gave two classes of counterexamples to deny the conjecture, and only one or two of the side loops contained in the two classes of bouquets were non-orientable. 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 side loop has an arbitrary number of non-orientable loops. The other is interpolating, whose side loop has an arbitrary number of both non-orientable loops and orientable loops.
Fig.1 Common segment (the segment where the vertex disk intersects with the edge disk), edge segment (the segment where the edge disk boundary removes the common segment), and vertex segment (the segment where the vertex disk boundary removes the common segment)
Fig.2 A ribbon graph with 2 boundary branches
Fig.3 Bouquet of signed rotation and its signed intersection graph
Fig.4 Bouquet C2n+a
Fig.5 Bouquet
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