Counterexamples to the interpolating conjecture on partial-dual genus polynomials of ribbon graphs

TL;DR

This paper provides counterexamples to a conjecture about the interpolating property of partial-dual Euler genus polynomials in ribbon graphs, challenging previous assumptions and expanding understanding of their behavior.

Contribution

The paper presents the first known counterexamples and infinite classes of counterexamples to the conjecture on partial-dual Euler genus polynomials in ribbon graphs.

Findings

Counterexamples disprove the conjecture.

Two infinite classes of counterexamples are identified.

Abstract

Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They showed that the partial-dual genus polynomial for an orientable ribbon graph is interpolating and gave an analogous conjecture: The partial-dual Euler-genus polynomial for any non-orientable ribbon graph is interpolating. In this paper, we first give some counterexamples to the conjecture. Then motivated by these counterexamples, we further find two infinite classes of counterexamples.