Parallel edges in ribbon graphs and interpolating behavior of partial-duality polynomials

TL;DR

This paper investigates the behavior of partial-duality polynomials in ribbon graphs with parallel edges, providing counterexamples to existing conjectures and exploring the conditions under which these polynomials are interpolating.

Contribution

It demonstrates that sufficiently parallel edges lead to negative answers to certain conjectures about partial-Petrial and partial-dual genus polynomials in ribbon graphs.

Findings

Presence of enough parallel edges makes the partial-Petrial polynomial not even-interpolating.

Counterexample shows that non-odd/non-even partial-dual genus polynomials can be interpolating.

Results challenge previous conjectures on the properties of partial-duality polynomials.

Abstract

Recently, Gross, Mansour and Tucker introduced the partial-twuality polynomials. In this paper, we find that when there are enough parallel edges, any multiple graph is a negative answer to the problem 8.7 in their paper [European J. Combin. 95 (2021), 103329]: Is the restricted-orientable partial-Petrial polynomial of an arbitrary ribbon graph even-interpolating? In addition, we also find a counterexample to the conjecture 8.1 of Gross, Mansour and Tucker: If the partial-dual genus polynomial is neither an odd nor an even polynomial, then it is interpolating.