On the interlace polynomials

TL;DR

This paper introduces a multivariate interlace polynomial extending previous interlace polynomials, capturing circuit partition generating functions for directed and undirected graphs, with invariance properties and recursive definitions.

Contribution

It presents a new multivariate interlace polynomial that unifies and extends existing interlace polynomials for directed and undirected graphs.

Findings

Defines a multivariate interlace polynomial for undirected 4-regular graphs.

Shows invariance under a refined local complementation.

Provides a simple recursive computation method.

Abstract

The generating function that records the sizes of directed circuit partitions of a connected 2-in, 2-out digraph D can be determined from the interlacement graph of D with respect to a directed Euler circuit; the same is true of the generating functions for other kinds of circuit partitions. The interlace polynomials of Arratia, Bollob\'as and Sorkin [J. Combin. Theory Ser. B 92 (2004) 199-233; Combinatorica 24 (2004) 567-584] extend the corresponding functions from interlacement graphs to arbitrary graphs. We introduce a multivariate interlace polynomial that is an analogous extension of a multivariate generating function for undirected circuit partitions of undirected 4-regular graphs. The multivariate polynomial incorporates several different interlace polynomials that have been studied by different authors, and its properties include invariance under a refined version of local complementation and a simple recursive definition.