Partial-duals for planar ribbon graphs

TL;DR

This paper studies the enumeration of partial-duals in planar ribbon graphs, providing formulas, recurrence relations, and asymptotic behavior insights, thereby advancing understanding of their genus distributions and challenging existing conjectures.

Contribution

It introduces a formula for maximum partial-dual genus, refutes a conjecture, and establishes recurrence relations and asymptotic normality for partial-dual genus distributions in planar ribbon graphs.

Findings

Derived a formula for maximum partial-dual genus.

Refuted the interpolating conjecture of Gross, Mansour and Tucker.

Proved asymptotic normality of partial-dual genus distributions.

Abstract

In 2009, Chmutov introduced the partial-duality for a ribbon graph $G$. Recently, Gross, Mansour and Tucker enumerated all possible partial-duals of $G$ by genus and introduced the partial-dual genus polynomial of a ribbon graph $G.$ This paper mainly enumerates partial-duals for planar ribbon graphs. First, we obtain a formula for the maximum partial-dual genus for any planar ribbon graph and give a negative answer to the interpolating conjecture of Gross, Mansour and Tucker. Then we show that there is a recurrence relation between the partial-dual genus polynomials of planar ribbon graphs $G-e$ and $G$. Furthermore, two related results are also given. These recurrence relations give new approaches to calculate the partial-genus dual polynomials for some planar ribbon graphs. In addition, we prove the asymptotic normality for some partial-dual genus distributions.