Binary nullity, Euler circuits and interlace polynomials

A theorem of Cohn and Lempel [J. Combin. Theory Ser. A 13 (1972), 83-89] gives an equality involving the number of directed circuits in a circuit partition of a 2-in, 2-out digraph and the GF (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 vertexnullity interlace polynomials of circle 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 a circle graph by attaching loops at some vertices then the vertex-nullity interlace polynomial qN (H) is essentially the generating function for certain circuit partitions of an associated 4-regular graph. 2000 Mathematics Subject Classification. 05C50