Partial-dual genus polynomial of graphs

TL;DR

This paper introduces a combinatorial approach to the partial-dual genus polynomial of graphs, extending its definition and proving it satisfies the four-term relation, addressing a problem posed by Chmutov.

Contribution

It provides a new combinatorial method for the partial-dual genus polynomial and extends its applicability to all graphs, confirming it satisfies the four-term relation.

Findings

The partial-dual genus polynomial can be defined combinatorially via intersection graphs.

The polynomial extends to all graphs beyond intersection graphs.

It satisfies the four-term relation of graphs.

Abstract

Recently, Chmutov introduced the partial duality of ribbon graphs, which can be regarded as a generalization of the classical Euler-Poincar\'e duality. The partial-dual genus polynomial $^\partial\varepsilon_G(z)$ 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.