Partial-dual genus polynomial of graphs

2025/02/26 by Cheng, Zhiyun
#05C10 #05C31 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2502.18950

Abstract

Recently, Chmutov introduced the partial duality of ribbon graphs, which can be regarded as a generalization of the classical Euler-Poincaré duality. The partial-dual genus polynomial ^∂ε_G(z) is an enumeration of the partial duals of G by Euler genus. For an intersection graph derived from a given chord diagram, the partial-dual genus polynomial can be defined by considering the ribbon graph associated to the chord diagram. In this paper, we provide a combinatorial approach to the partial-dual genus polynomial in terms of intersection graphs without referring to chord diagrams. After extending the definition of the partial-dual genus polynomial from intersection graphs to all graphs, we prove that it satisfies the four-term relation of graphs. This provides an answer to a problem proposed by Chmutov.

Related

• Partial-dual genus polynomials and signed intersection graphs
• Counterexamples to conjectures by Gross, Mansour and Tucker on partial-dual genus polynomials of ribbon graphs
• Partial-duals for planar ribbon graphs
• A categorification for the partial-dual genus polynomial
• Counterexamples to the interpolating conjecture on partial-dual genus polynomials of ribbon graphs