On the interlace polynomials
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.
Cite
@article{arxiv.1008.0091,
title = {On the interlace polynomials},
author = {Lorenzo Traldi},
journal= {arXiv preprint arXiv:1008.0091},
year = {2012}
}
Comments
v7: 37 pages, 10 figures. Many corrections and improvements have been made since the first version was posted. Further changes may be made before publication in JCTB