Binary nullity, Euler circuits and interlace polynomials

A theorem of Cohn and Lempel [M. Cohn, A. Lempel, Cycle decomposition by disjoint transpositions, J. Combin. Theory Ser. A 13 (1972) 83–89] gives an equality relating the number of circuits in a directed circuit partition of a 2-in, 2-out digraph to the G F ( 2 ) -nullity of an associated matrix. This equality is essentially equivalent to the relationship between directed circuit partitions of 2-in, 2-out digraphs and vertex-nullity interlace polynomials of interlace graphs. We present an extension of the Cohn–Lempel equality that describes arbitrary circuit partitions in (undirected) 4-regular graphs. The extended equality incorporates topological results that have been of use in knot theory, and it implies that if H is obtained from an interlace graph by attaching loops at some vertices then the vertex-nullity interlace polynomial q N ( H ) is essentially the generating function for certain circuit partitions of an associated 4-regular graph.