Mathematics – Combinatorics
Scientific paper
2010-07-31
Mathematics
Combinatorics
v1: 26 pages, 9 figures; v2: 27 pages, 9 figures; v3 and v4: 30 pages, 10 figures; v5: 36 pages, 10 figures. v2 and v3 incorpo
Scientific paper
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.
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