2022/04/17 by Qiyao Chen, Chen, Qiyao, Yichao Chen +1
Mathematics · #Advanced Combinatorial Mathematics #Finite Group Theory Research #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2204.08010
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.