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 [R. Arratia, B. Bollobás, G.B. Sorkin, The interlace polynomial of a graph, J. Combin. Theory Ser. B 92 (2004) 199–233; R. Arratia, B. Bollobás, G.B. Sorkin, A two-variable interlace polynomial, 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.