2010/07/31 by Lorenzo Traldi, Traldi, Lorenzo
Mathematics · #05C50 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C50
paper · pdf · doi:10.48550/arxiv.1008.0091
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
arxiv created 2012/09/20 · arxiv updated 2012/09/24
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 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.