Twist polynomials of delta-matroids

TL;DR

This paper introduces twist polynomials for delta-matroids, extending concepts from ribbon graphs, and characterizes their properties, including conditions for having only one term and the relation to bipartite intersection graphs.

Contribution

It defines twist polynomials for delta-matroids and explores their properties, providing characterizations and linking them to intersection graph bipartiteness.

Findings

Twist polynomials generalize partial duality polynomials to delta-matroids.

Characterization of even normal binary delta-matroids with single-term twist polynomials.

Twist polynomial's constant term relates to the bipartiteness of the intersection graph.

Abstract

Recently, Gross, Mansour and Tucker introduced the partial duality polynomial of a ribbon graph and posed a conjecture that there is no orientable ribbon graph whose partial duality polynomial has only one non-constant term. We found an infinite family of counterexamples for the conjecture and showed that essentially these are the only counterexamples. This is also obtained independently by Chumutov and Vignes-Tourneret and they posed a problem: it would be interesting to know whether the partial duality polynomial and the related conjectures would make sence for general delta-matroids. In this paper, we show that partial duality polynomials have delta-matroid analogues. We introduce the twist polynomials of delta-matroids and discuss its basic properties for delta-matroids. We give a characterization of even normal binary delta-matroids whose twist polynomials have only one term and then prove that the twist polynomial of a normal binary delta-matroid contains non-zero constant term if and only if its intersection graph is bipartite.