On the interlace polynomials

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, Bollobás 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 di¤erent interlace polynomials that have been studied by di¤erent authors, and its properties include invariance under a re…ned version of local complementation and a simple recursive de…nition. Keywords. circuit partition, interlace polynomial, isotropic system, local complementation, pivoting, split graph Mathematics Subject Classi…cation. 05C50